Logic in practice

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General Editor : G. C. Field, M.A., B.Sc.

LOGIC IN PRACTICE

METHUEN’S MONOGRAPHS ON PHILOSOPHY
AND PSYCHOLOGY

Fcap. 8vo, 2s. 6d. net each.

General Editor : G. C. Field, M.A., B.Sc., Professor of
Philosophy in the University of Bristol.

PREJUDICE AND IMPARTIALITY. By G. C. Field,
M.A., B.Sc.

THE PSYCHOLOGY OF STUDY. By C. A. Mace, M.A.,
Lecturer in Psychology in the University of St. Andrews.

THE EVIDENCE OF OUR SENSES. By A. W. P.
Wolters, M.A., Lecturer in Psychology and Education,
University of Reading.

INTELLIGENCE AND INTELLIGENCE TESTS. By
Rex Knight, M.A., Anderson Lecturer in Comparative
Psychology in the University of Aberdeen.

PSYCHOLOGY AND THE CHOICE OF A CAREER.
By F. M. Earle, M.Ed., D.Sc., formerly Head of the
Vocational Department of the National Institute of
Industrial Psychology.

LOGIC IN PRACTICE. By L. S. Stebbing, D.Lit.,
Professor of Philosophy in the University of London.

In Preparation

THE BACKWARD SCHOLAR. By A. Macrae.

LOGIC IN PRACTICE

by

L. SUSAN STEBBING

D.LIT., M.A.

PROFESSOR OF PHILOSOPHY IN THE UNIVERSITY OF LONDON

METHUEN & CO. LTD.

36 ESSEX STREET W. C.
LONDON

First Published in 1934

WELLCOME INSTHJT!
LIBRARY [

CoH.

WelM0r3C

Coll.

I

No.

2_

PRINTED IN GREAT BRITAIN

TO

MY SISTERS

‘ No book can do ALL a man’s thinking for him. The
utility of any statement is limited by the willingness of
the receiver to think.’ — Ezra Pound

PREFACE

HAT passes for knowledge in ordinary life is

V V most often nothing but beliefs which we hold
more or less tenaciously without any clear awareness
as to what precisely we are claiming to know. Even
when our beliefs happen to be true, as is sometimes
the case, our lack of precision and our ignorance of
the grounds upon which these beliefs could be based
permit us to hold other beliefs which are contra-
dictory. We are apt to be cocksure where we should
be hesitant, to be vague where precision is important,
and to be contentious although argument is possible.
These defects are obvious in the case of other people’s
assertions ; we must be exceptionally fortunate — or
unusually stupid — if we have never noticed them in
our own thinking. It must be the desire of every
reasonable person to know how to justify a contention
which is of sufficient importance to be seriously
questioned. The explicit formulation of the principles
of sound reasoning is the concern of Logic.

The study of logic does not in itself suffice to enable
us to reason correctly, still less to think clearly where
our passionate beliefs are concerned. Thinking is
an activity of the whole personality. Given, however,
a desire to be reasonable, then a knowledge of the
conditions to which all sound thinking must conform
will enable us to avoid certain mistakes into which
we are prone to fall. There is such a thing as a habit
of sound reasoning. This habit may be acquired by
consciously attending to the logical principles of

vii

Vlll

LOGIC IN PRACTICE

sound reasoning, in order to apply them to test the
soundness of particular arguments. No doubt there
are a few gifted persons whose critical temper of mind
enables them to reason soundly although they have
never had occasion to attend to the principles in
accordance with which their reasoning proceeds.
There may be others too stupid ever to be able to
appreciate the logical force of an argument. Most
people, however, are between these extremes. Their
reasons are sometimes sound, sometimes unsound,
but they often do not know why they are the one or
the other. It is for such people that this book is
intended.

It has been impossible in so small a book to do
more than touch upon many topics which are worth
detailed consideration. Technicalities have been
deliberately avoided, for this book is in no sense
intended to provide an introduction to Logic. Stress
is laid upon the importance of considering language,
which is an instrument of our thinking and is imper-
fect, as are all human creations.

I desire to thank Mr. A. F. Dawn and Miss J. Wynn
Reeves for their help in the correction of the proofs,
and I am further indebted to the latter for her help
in the compilation of the index.

L. S. S.

Bedford College

University of London
December 1933

CONTENTS

CHAPTER PAGE

PREFACE Vii

I. PURPOSIVE THINKING 1

n. THE IMPORTANCE OF FORM . . .13

III. DEDUCTIVE FORMS 32

IV. AMBIGUITY, INDEFINITENESS, AND RELE-

VANCE 64

V. THE ESTIMATION OF EVIDENCE ... 85

VI. THE GROUNDS OF OUR BELIEFS ... 96

REFERENCES FOR READING .... 109

INDEX Ill

IX

LOGIC IN PRACTICE

CHAPTER I

PURPOSIVE THINKING

‘ Where the senses fail us reason must step in.’ — Galileo

HINKING is an activity ; we think in order to

JL do. But not all doing consists in overt action
producing perceptible changes in the given situation
or environment. The ‘ man of action ’ is commonly
opposed to the ‘ man of thought ’. There are good
grounds for this opposition ; but even men of action
have to think, however much their activities may
suggest the contrary. The world to-day needs clear
thinkers even more than it needs men of good will,
and not less than it needs men of great practical
energy. To be confronted with a problem is to be
compelled to think. Thinking essentially consists in
asking questions and attempting to answer them. To
ask a question is to be conscious of a problem ; to
answer correctly is to have discovered its solution.
Purposive thinking is thinking directed to answering
a question held steadily in view. Such directed
thinking may be contrasted with idle reverie.1

Suppose a man lying awake in his cabin on board
a passenger steamer. He listens to the sound of the
sea, to the numerous slight sounds — the creakings
and strainings — always audible on board ship. His

1 Cf. C. A. Mace : The Psychology of Study, Chap. IV. In this
chapter will be found an excellent account of the process of
thinking, treated from the point of view of the psychologist.

i 1

o

LOGIC IN PRACTICE

hearing of these sounds may partly determine the flow
of his thoughts ; he passes idly from one thought to
another. Suddenly he hears a loud, distinctive sound
- — the three long booms which are the danger-signal.
This sense-impression is significant ; he does not
notice it merely as a sound ; it signifies for him — ship
in danger. He springs up, snatches a coat, and
rushing out hears the word ‘ Fire ! ’ The reader's
imagination may supply the details. Provided that
the man be not too panic-stricken to think at all, his
thinking will now be purposive ; it will be directed
to securing his own safety or that of others. He will
now actively connect one apprehended fact with
another. Once the fire-situation is grasped, his thinking
will be directed to a practical end ; the conditions of
attaining this practical end will constitute the problem
which his thinking is directed to solving.

Suppose now that a committee of investigation is
confronted with the problem of how the fire originated.
This problem is purely theoretical, however much the
desire to solve it may be the practical desire of
assessing the responsibility for the outbreak of fire,
or of attempting to prevent the occurrence of such
accidents in the future. A problem is not made a
practical problem simply because its solution may
have practical applications. The committee are
seeking to obtain knowledge ; they want to find a
true answer to a definite question. Their problem is
as purely theoretical as the problem of determining
the conditions of combustion in general, or the prob-
lem of determining the nature of eclipses. The
distinction between what is often called practical
thinking and theoretical thinking lies wholly in the
purpose for which the thinking is pursued. In both

PURPOSIVE THINKING

3

cases the thinking process is the same ; it is purposive,
and thus directed. The opposition is not between
practical and theoretical thinking but between
directed thinking and idle reverie.

Intelligent dealing with a problem involves, first,
the apprehension of a situation giving rise to the
problem ; secondly, the explicit awareness of the
question constituting the initial stage of the problem ;
thirdly, formulation of the conditions to which the
solution must conform. These conditions are deter-
mined by the total situation. In so far as these
conditions are clearly apprehended and selectively
attended to, precise questions can be formulated and
various answers tried out. The point to be stressed
is that to ask an intelligent question is to have taken
note of the conditions set by the problem ; to suggest
an intelligent answer is to have discerned within the
situation, so far as apprehended, those factors which
may be relevant to the solution. Intelligent answers
may be wrong, but they are never off the point. To
keep to the point is to be guided by relevant con-
siderations alone.

The importance of excluding irrelevant suggestions
cannot be overstressed. In the case of the man on
the burning ship, it is clear that his thinking would
be effective only in so far as what came into his mind
— or, at least, was taken note of — was relevant to the
conditions constituting his problem. If he were to
consider whether he could fly like a bird from the
ship, or whether the flames could be extinguished by
a heavy rainfall, he would be asking questions the
answers to which could have no bearing upon the
difficulty with which he was confronted. In the same
way the committee investigating the cause of the fire

4 ’ LOGIC IN PRACTICE '

would be making no progress towards the solution ot
their problem if they were to ask whether the fire
was due to the ship’s having sailed from port on a
Friday, or whether it was caused by a hot dispute
between members of the crew, or whether it was due
to the wrath of God because the passengers were
dancing on a Sunday. That these suggestions at once
strike the reader of this book as absurd is due to his
knowing too much about the kind of situation, and
thus about the conditions of the problem under dis-
cussion, even to entertain the possibility that such
factors could be relevant. It might well be relevant,
however, for the committee to ask whether the fire
was due to the careless dropping of a lighted match
or of an unextinguished cigarette end, or to the fusing
of an electric wire, or to a deliberate act of incen-
diarism. Each of these questions would suggest other
questions the answers to which might more easily be
ascertained to be correct or incorrect. In this way
progress towards a solution may be made. We may
briefly consider each of the last three suggestions in
order to see how a relevant answer may admit of
testing.

The question the committee sought to answer was
quite definite, namely, What caused this fire ? They
sought to discover, not a possible cause of fire on board
ship, but the actual cause of this fire. Each suggestion
presents a possible cause. Other possible causes
might have been considered. The only way to test
these suggestions is to ask what else would have
happened if the given suggestion were correct, but
which would not have happened if one of the other
suggestions were correct. To be able to ask these
further questions the committee would need to know

PURPOSIVE THINKING

5

a good deal about the ship and about what took place
when the fire occurred ; they would also have to
possess certain technical knowledge. If the ship had
been completely burnt out, relevant questions could
not be answered. If, however, the burning ship had
been towed into port, if there were survivors who
could answer definite questions, if the committee
possessed other relevant knowledge, it might be
possible to say what was most probably the cause.
These conditions may be assumed to be fulfilled, for
without some means of obtaining such information
there would be nothing to investigate.

Each of the suggestions put forward is a supposition
to the effect that some unobserved occurrence happened,
this occurrence being such that, if it had happened,
then fire would have broken out. A supposition thus
entertained in order to account for what happens is
called an hypothesis. The committee seriously con-
sidered three hypotheses. Each hypothesis has certain
consequences , i.e. given that the hypothesis were cor-
rect something else would have happened. The
question then arises whether it did happen.

(1) The hypothesis that the fire was caused by a
carelessly dropped lighted match suggests the follow-
ing questions : (a) Did the fire begin in a cabin or
in a public part of the deck ? ( b ) Did it occur at
night ? If the answer to ( b ) is affirmative, and if the
answer to (a) is that the fire broke out in the luggage-
room, then it is improbable that this first hypothesis
is correct, since it is unlikely that any one would be
smoking in the luggage-room, especially at night.

(2) The hypothesis that the cause was the fusing of
an electric wire suggests the questions : (c) Did the
fire spread along the electric cables ? (d) Had the

6

LOGIC IN PRACTICE

electric installation of the ship been recently over-
hauled so that it was known to be in good repair ?
(e) Were some parts of the ship in darkness whilst
in other parts the switches worked ? An affirmative
answer to (c) would not cut out the possibility of the
lighted match as the cause, provided that the fire had
broken out in a cabin, and provided that the cabins
had been treated with non-inflammable varnish. If,
however, the fire had broken out at night and in a
little-frequented part of the ship, then its spreading
along the electric cables tells in favour of this second
hypothesis. But if the answer to ( d ) were ‘ Yes ’, and
to (e) were ‘ No ’, then the available evidence is not
in favour of the fused wire as the cause. At this stage
the third hypothesis seems plausible, namely, that
some one deliberately set the ship on fire.

(3) This hypothesis is not one which would readily
be accepted. Careless smokers and fused wires are
familiar enough. Certainly incendiarists are not un-
known. But no sane, ordinary passenger would put
himself to the inconvenience, perhaps even peril, of
being on a burning ship. Thus, unless the incendiarist
were a madman, he must have had some strong motive
for so acting. This hypothesis suggests the question
whether any one on board could have expected to
derive some benefit from the destruction of the ship,
or was the agent of some one who had such expecta-
tions. In attempting to answer this question the
committee would be led to pursue investigations
beyond the occurrences on the voyage itself. They
would ask what benefit could be derived from the
destruction of the ship, and who would be benefited.
Further questions at once arise. (/) Was the ship
heavily insured ? ( g ) What was the age of the ship ?

PURPOSIVE THINKING

7

( h ) Were the owners in need of money ? Suppose it
were found that the ship was heavily insured ; that
its future sea -going life was not likely to be long ;
that the owners needed the insurance money ; then
this third hypothesis is worth taking seriously. If it
were further discovered that other ships belonging to
the same Line had been recently destroyed by fire,
then the hypothesis will seem not unreasonable under
the circumstances.

It is not worth while to pursue this illustration
further. Enough has been said to show how an
intelligent person confronted with a problem will
proceed to ask questions and guess at the answer ;
how various answers lead to other questions and
further guesses. A guess is worth making only if the
answer can be tested. These guesses are suggestions
as to what may possibly be the case ; i.e. they are
hypotheses. These hypotheses are worth entertaining
only if the possibilities can be narrowed down. The
possibilities can be narrowed down by discovering
that what would be possible in certain other circum-
stances is not possible under the ascertained circum-
stances. In the next chapter we shall see how such
an investigation conforms to certain principles which
interest the practical logician.

Thinking out a problem involves inference.
Inference is a mental process in which a thinker passes
from something given or taken for granted — the
datum — to something which he accepts because, and
only because, he has accepted the datum. It is a
passage of thought from datum to conclusion. To
accept a conclusion as the result of an inference is to
accept it upon the basis of what is taken to be evi-
dence. To regard what is observed, or is believed, or

8

LOGIC IN PRACTICE

is in any way apprehended, as evidence is to regard it
as indicating something other than itself. To regard
a fact as providing evidence is, then, to regard that
fact as significant of some other fact. We may have
some evidence in favour of a conclusion and no
evidence contrary to it, and yet the evidence may not
be decisive. Evidence is decisive of a certain con-
clusion when the acceptance of the evidence necessi-
tates the acceptance of the conclusion. Unfortunately
we may be mistaken both with regard to what the
facts are and with regard to what the facts indicate.

The example of the committee’s investigation has
shown us that intelligent guessing is controlled by the
recognition of certain ascertained conditions as rele-
vant to the solution of the problem. These conditions
relate to matters of fact. They may be called the
material conditions of the problem since they are
provided by the special subject-matter of the investi-
gation. To apprehend a condition as relevant to the
solution is already to know a certain amount about
the nature of situations resembling the given one in
certain respects. If we knew all about the situation
there would be no problem. If we knew nothing about
similar situations we should not be able even to ask
intelligent questions about it. Relevance is signifi-
cance for the question at issue. Nothing is significant
in itself. That a ship is heavily insured acquires
significance if the ship belongs to owners in financial
difficulties and if its destruction occurred in a manner
compatible with a deliberate act of incendiarism ; it
ceases to be significant if the owners are known to be
men of incorruptible honesty. A red light at a street
corner signifies ‘ stop ’ to a motorist who knows the
conventions of traffic regulation. A certain -shaped

PURPOSIVE THINKING

9

mark on the sand is significant of the previous
presence of a human being only to one who can inter-
pret the mark as a footprint. A grey mark on a carpet
may be significant of a certain brand of cigars to one
who, like Sherlock Holmes, has made a special study
of varieties of cigar ash. To multiply examples is
unnecessary. The point must be, obvious to any one
on reflection. Nevertheless, we are apt to forget that
significance arises only in so far as a given fact
indicates something other than itself. The possibility
of such indication depends upon knowledge possessed
by the person apprehending the significance. The
detective who 4 takes in the whole situation at a
glance ’ would need already to know what each item
he observes signifies. In sober fife this is not the case.
The observed facts acquire significance only when
viewed in the light of a definite question which guides
his thinking. Reviewed in the light of a different
question, the same facts may alter in significance.
Readers familiar with detective novels will be able to
supply examples illustrating this point.

It should now be clear that significance is a property
of signs. A sign indicates something other than itself.
Signifying is a relation requiring three terms : a sign,
that of which the sign is significant, and a thinker
for whom the former indicates the latter. Just as a
book, for example, is not a gift unless it be given by
some one to some one, so a red light, for example, is
not a sign unless it be interpreted by some one to
indicate something. To be able thus to interpret a
presented fact we must know something not given in
the presented situation.

It is a mistake to regard thinking which involves
interpretation of signs as confined to the higher levels

10

LOGIC IN PRACTICE

of consciousness, or to confine problems to what are
often called intellectual problems. There are no hard
and fast distinctions in the development of mental
life. The distinction between practical and theoretical
problems does not relate to the kind of thinking
involved, but to the set of purposes within which the
problem originates, and hence to the kind of changes
its solution is designed to bring about. A problem
may be practical in the sense that it relates to bringing
about a change in the environment, for example,
making a box, or mending a clock, or disposing an
army, or winning an election campaign. A problem
may be theoretical in that it relates to bringing about
a change in the mind of a thinker who has no other
purpose in view than to apprehend a situation more
clearly by discerning the connexion between its com-
ponent elements. The purpose of solving the problem
is, in the latter case, the acquirement of knowledge ;
in the former, the doing of something. In both alike,
thinking is directed to an end determined by the
nature of the problem. In effective thinking con-
sciousness of the conditions set by the problem directs
the cognitive activity of the thinker, determining
what shall come to mind.

Directed thinking in its most highly developed form
is reasoning. To be reasonable is to be capable of
apprehending a situation as a whole, to take note of
those items which are relevantly connected, and not
to connect arbitrarily items not apprehended as signi-
ficant. Apprehension of relevance depends upon two
quite different factors : knowledge and sagacity. ‘ To
be sagacious,’ William James has said, ‘ is to be a
good observer.’ This statement may be accepted if
we admit that a good observer is one capable of

PURPOSIVE THINKING

11

discerning relevant connexions. Certainly a good
observer is not one who ‘ stands and stares Staring
is a sign of stupidity, or of an astonishment so pro-
found as temporarily to destroy the power to think
reflectively. In reasoning we select from a set of items
presented to us just those which are significant of
the facts we are trying to establish. Failure to be
reasonable may occur in one or other of two very
different ways. Significance may be missed through
sheer ignorance or through passion and emotional
attitudes which check clear-sighted apprehension. No
doubt we all desire to be reasonable but few of us
achieve this desire. Even those few are reasonable
only at rare moments. Ignorance and passion present
grave obstacles to be overcome only by a supreme
desire to think clearly. The scientist is regarded as
the exemplar of rationality. Within the field of his
expert investigations he is, for here he has both
relevant knowledge and a disinterested desire to
understand merely for the sake of understanding.
Confronted with problems of a different kind, such as
those presented by a political crisis or by the behaviour
of his children, he may allow passion to subdue
reasoning, stupidly asking irrelevant questions and
accepting irrelevant answers.

Correct answers are as rare as they are difficult.
Human beings from infancy upwards are confronted
with problems. Our ability to deal with these prob-
lems effectively depends in part upon our ability to
think clbarly. It is as natural for men to think as to
walk and to speak. Few of us, however, walk or
speak well, although we may remain for ever unaware
of our deficiencies in this respect. Where few attain
a high level of excellence, the majority are apt not to

12

LOGIC IN PRACTICE

notice that their performance falls short. Our natural
capacity to walk or to speak may be improved by
practice, guided by some standard which we can try
to reach. The case is not otherwise with thinking.
Although we all must think, we seldom think effec-
tively. Our thinking is more likely to be effective if
we are aware of the conditions to which efficient
thinking must conform. To know these conditions is
to have a standard by reference to which we may
gauge the success of our efforts. In this way we may
learn to avoid some mistakes.

CHAPTER II

THE IMPORTANCE OF FORM

* Importance depends on purpose and on point of view.’

A. N. Whitehead

HE problems about which we can think effec-

X tively arise out of situations which are on the
whole familiar. Were this not so we should not know
what questions to ask, still less what answers were
relevant. In this book we are not concerned to
inquire how we come to have such familiarity. We
are not asking how knowledge originates. Our
problem is limited to considering how we may reason-
ably come to accept conclusions we did not know
before and now accept because, and only because, we
have accepted something else. Knowledge thus
obtained is obtained by inference. It is not always
easy to draw the line between a judgment in which
no inference is involved and an inferred conclusion.
In judging ‘ That is a cow * Here is a pen ‘ That
is a motor passing down the street ’, we are merely
recording what we take to be directly given. It is
true that such judgments go beyond what is directly
given to sense, but it does not follow that they are
reached by inference. If they are questioned we may
attempt to justify them by inference from something
indubitably given. Certainly such judgments may
be erroneous. Nevertheless, recognition, perception,
memory, provide us with the materials from which
we start. None of these is completely trustworthy,
but they are all we have.

13

14

LOGIC IN PRACTICE

The passenger on the ship — in the illustration
considered in the last chapter — recognized a certain
sound as a danger-signal. Had he not so recognized
the sound, there would have been no problem for him.
The committee of investigation assumed that happen-
ings do not occur haphazardly, but are so connected
that what happens here — now is conditioned by what
happened there — then. In their inquiry they took for
granted a number of generalizations with regard to
the way things of a certain sort behave in relation to
other things, e.g. lighted matches in relation to wood,
careless smokers in relation to lighted matches, fused
wires in relation to electric cables. Things of a certain
sort constitute a class. The phrase ‘ of a certain sort ;
indicates that the things possess in common, properties
which do not all belong to any other thing. Class-
names, or common nouns, are used to refer to all the
things of a certain sort. The class-name ‘ ship ’, for
example, is used to enable us to refer to many objects
each of which behaves in the same sort of way in
relation to other things. No two ships have all their
properties in common, but the differences between
one ship and another can often be ignored as irrelevant
to what we are thinking about. We may, however,
wish to take note of some of these differences. If we
are in a hurry to cross the Atlantic Ocean it matters
to us whether the ship is propelled by steam or by
wind, whether it has efficient engines, and so on. Thus
we distinguish between ‘ steam ships ’ and ‘ sailing
ships ’, between ‘ screw steamers ’ and ‘ turbine
steamers ’, between ‘ sloops * schooners ‘ cutters ’,
and ‘ brigs ’. A ship can also be regarded from the
point of view of its tonnage, or of its capacity for
holding so many passengers, or of its belonging to a

THE IMPORTANCE OF FORM

15

certain Line, or of its being heavily insured. In
taldng note of these different properties we put a ship
into different classes. A given definite object is a
member of many different classes. A class is nothing
but a set of objects possessing in common properties
not all of which are possessed by any object not
belonging to that class. Whenever the common
possession of these properties is important for our
purposes we use a class-name. When we want to
distinguish from other objects a set of objects possess-
ing in common a property not previously noted, we in-
vent a class-name, e.g. ‘ X-rays ‘ Bolshevik ’, ‘ taxi
The use of class-names enables us to economize
thought. It would be inconvenient if we could not
briefly refer to all those vehicles which we now distin-
guish from other vehicles, such as horse-cabs. What
we, in England, understand by ‘ taxicab ’, and its
familiar abbreviation, ‘ taxi ’, is not distinguished
merely by the possession of a taximeter, so that the
Englishman in Italy may be surprised to find a
horse-drawn vehicle answer his summons for ‘ a taxi ’.
How we use a word, i.e. what we mean to refer to by
using it, is largely determined by the situations in
which we have met the objects to which the word
refers. We shall see later that herein lies a certain
danger to clear thinking in the use of familiar class-
names.1 At present we are concerned only to notice
how indispensable is this convenience of language.
Class-names enable us to abbreviate and to connect.
The psychological reasons which make the invention
of such words as ‘ taxicab ’ and ‘ Bolshevik ’ useful
also lead to the specially devised terminology of a
special science. The chemist finds it useful to speak

1 See Chapter TV.

16

LOGIC IN PRACTICE

of ‘ carbon monoxide ’ and ‘ carbon dioxide and
even to abbreviate further by using the carefully
devised symbolism CO and C02. Formerly (about
1630) carbon dioxide was called ‘ gas sylvestre ’ ; later
(about 1754) it was called ‘ fixed air Each of these
names is significant ; the former of its discoverer, the
latter of one of its properties. The name £ carbon
dioxide ’ is, however, more importantly significant to
the chemist in showing the way in which this gas is
composed. The reader will be able to think of many
other examples of such convenient and economical
procedure. Words used in ordinary situations are not
so significantly devised. They are used to indicate
the presence of characteristics, or properties, wfiich
we have frequently found to be possessed by an object
in various situations. The point to be stressed is that
in using the common nouns which belong to our
everyday vocabulary we are benefiting by knowledge
acquired throughout the course of human history.
A class-name refers to important properties.

Throughout the preceding discussion the word
‘ important ’ has been frequently used, and the
corresponding abstract noun appears in the title of
this chapter. It is desirable to understand what the
word ‘ important ’ means. To say of anything that
it is ‘ important ’ is to say that it ‘ makes a difference
relevant to our purpose ’. Nothing has importance
apart from a purposive being. What makes a
difference for one purpose makes no difference for
another purpose. Hence, importance is relative to a
point of view. It must be insisted that for different
purposes different properties are important. The
reader would be well advised if, whenever he meets
the word ‘ important ’ in a serious discussion, he were

THE IMPORTANCE OF FORM

17

to ask himself : * Relative to what purpose, or from
what point of view, is this important ? ’ A few
examples may make this point clear.

Men are alike in certain respects ; they differ in
others. Similarly, with cows, ships, rainbows, novelists,
conservatives, or any other familiar class the reader
can think of. Although men differ from cows in
certain respects, they are like them in others. The
respects in which men are alike are important ; hence
we have the class-name. The differences between a
man and a cow are important ; we therefore refer to
them by different class-names. But since men and
cows are alike in that both are animals, we have
another class-name, ‘ animal ’, connecting them into
a wider class. Animals differ in important respects
from ships ; they are like them in having weight and
being movable. ‘ Important ’ has here been used to
indicate ‘ making a difference to the ordinary purposes
of ordinary men ’. So important for these purposes
are the differences between the classes mentioned, that
the grouping together of the objects in each class,
and the distinction of the class thus formed from
other classes, forms part of our everyday knowledge.
The various likenesses and differences are obvious in
any situation ; they cannot fail to strike our atten-
tion. Sometimes, however, an unobvious likeness
may be more important than a striking difference.
A good example is given by Bain, who points out that
‘ we become oblivious of the difference between a
horse, a steam engine, and a waterfall, when our
minds are engrossed with the one circumstance of
moving power \x To single out the characteristic of

1 The Senses and the Intellect, p. 521. Cf. L. S. Stobbiug :
A Modern Introduction to Logic, pp. 6-7.

18

LOGIC IN PRACTICE

moving 'power involves the imaginative selection of one
character out of a complex situation in order that it
may be attended to in isolation. In so doing we
ignore differences which, from another point of view,
would be important. Thus we analyse a situation and
abstract characters discernible within it. This process
of analysis and abstraction is involved in finding a
hidden face in a jmzzle-picture. On the basis of past
experience what is first apprehended as a medley of
tangled lines may come to be seen as a sketch of a
man smoking.1 In such apprehension abstraction,
analysis, and subsequent synthesis are involved.

What we apprehend, then, is conditioned by what
we have previously apprehended. This should have
been made clear by our discussion of significance in
the preceding chapter. For the purposes of inference
an important property is a property which can be
taken as indicating the presence of another propert}^.
In our everyday experiences we do find properties
constantly so conjoined that we can infer from the
presence of one to the presence of another. Since
things of the same sort exhibit characteristic modes
of behaviour in determinate situations, the recognition
of an object as belonging to a certain class may enable
us to discover how it will behave, or how it has
behaved, on some unobserved occasion. We say
Wood burns easily, meaning thereby to assert that
every piece of wood will so behave in the presence of
fire ; we are not referring only to those pieces of wood
which have been observed to burn. This characteristic
mode of behaviour on the part of wood is its property
of being inflammable. If we say The broadcast speeches

1 Cf. A. W. P. Wolters : The Evidence of our Senses (Methuen’s
‘ Monographs on Philosophy and Psychology ’), p. 41.

THE IMPORTANCE OF FORM

19

of 'politicians do not express the views they really hold,
we are saying something about the way politicians
behave in the situation of broadcasting speeches.
Such statements are made about the whole of a class
on the ground of the observed characteristics of some
of its members. A statement of this kind is an
empirical generalization. The use of a class-name
itself results from generalization, since, in applying
the class-name to an object we are asserting that the
object possesses properties not directly observed and
belonging also to other objects, although we may not
be aware that we are doing so until the application
of the class-name is challenged. Empirical generaliza-
tions may be false ; class-names may be 'wrongly
applied. As practical logicians we need to ask under
what conditions we may reasonably rely upon
empirical generalizations and may safely use class-
names. It is the purpose of this book to afford some
help towards answering these questions.

The kind of inference involved in reaching generali-
zations must be distinguished from the kind of
inference exemplified in applying generalizations to
particular cases. In this chapter we shall be mainly
concerned with the latter, but a few words may be
said about the former kind of inference.

Generalization involves a general property, i.e. a
property which may belong to many things. Having
noticed that all the things of a certain sort, which
have been observed, behave in certain ways, we infer
that any other thing of that sort, although it has not
been observed, will behave in that way. For example,
if, on the ground that all the psychoanalysts we have
met have been deficient in humour, we conclude that
every psychoanalyst lacks humour, we are generalizing

20 LOGIC IN PRACTICE

from some members of a class to all its members. This
is an example of what is known as inductive inference.
The observed instances constitute the datum of the
inference. The datum provides the premiss of the
inferred conclusion. We should not claim that those
psychoanalysts we happen to have met are all the
ones there are ; indeed, we want to assert that
deficiency in humour is characteristic of them as a
class, containing observed and unobserved members.
It may well be that the ones we have met were an
unfortunate selection ; they may not have been
representative. A single exception contradicts the
generalization. It is the characteristic of inductive
inference that the premisses may be true and yet the
conclusion may be false. This is possible because
inductive inference goes beyond the evidence.

Let us suppose that the application of a class-name
has been challenged. How do we proceed to answer
the challenge ? We seek to point to some charac-
teristic which belongs to every member of that class
and to nothing else. Let us imagine two people
looking out to sea. A says, ‘ That’s a cutter.’ B
replies, ‘ No ; it is a sloop.’ Here A’s application of
the class-name ‘ cutter ’ has been challenged. In
replying to the challenge A will make use of previously
acquired knowledge. He may answer as follows : ‘ It
is a cutter, for not only is it single-masted, but it has
a running bowsprit and no jib-stay.’ A has now
presented his original statement as the conclusion of
an inference ; he has supported the statement by
producing reasons. In thus reasoning A has made use
of previous knowledge, involving generalizations, to
suggest characteristics the possession of which affords
a test of the correctness of the original statement.

THE IMPORTANCE OF FORM

21

Since we are not interested in the mythical dispute
between A and B, but in the nature of A’s reasoning,
we may proceed to set out the steps of his reasoning
at a length which people engaged in the pleasant
pastime of watching ships coming into harbour could
only regard as tedious and pedantic. The steps may
be set forth as follows :

Alternatives : It is a cutter or a sloop.

Suggested test : Has it a running bowsprit and a
jib-stay ?

Argument : No sloops have running bowsprits and
jib-stays ;

This ship has a running bowsprit and a jib-stay ;

This ship is not a sloop.

All cutters have running bowsprits and a jib-stay ;

This ship has a running bowsprit and a jib-stay ;

This ship is perhaps a cutter.

But (it was agreed) it is either a cutter or a sloop ;

And (it has been shown) it is not a sloop ;

This ship is a cutter.

This reasoning is an example of deductive inference.
The reasons offered are the premisses of the inference.
These premisses are taken to be true, and it is shown
that, this being so, the conclusion must be accepted.
To say that we must accept a conclusion is to say that
we should not be rational in accepting the premisses
and rejecting the conclusion. This is the distin-
guishing characteristic of deduction. It would be
quite reasonable to accept the premiss, Some psycho-
analysts lack humour whilst admitting that possibly
not all do. The formal distinction between inductive
and deductive inference consists in the fact that the
conclusion of an inductive inference may be false

22

LOGIC IN PRACTICE

although the premisses are true ; whereas, the con-
clusion of a correct deductive inference cannot be
false provided that the premisses are true. Hence, in
deduction the truth of the premisses is a guarantee of
the truth of the conclusion.

It should be observed that we have said 1 provided
that the premisses are true ’. The truth or falsity of
the premisses is determined by their relation to facts,
i.e. to definite states of affairs which are the case.
This relation of premiss to fact is what, in Chapter I,
we called the material conditions of a problem. It
may be the case that it is false that no sloops have a
running bowsprit and a jib -stay ; in the argument,
however, it was taken for granted that the premiss
was true. We cannot (as we shall see later) establish
its truth by deductive inference. We can only assert
that if it is true, then something else must also be
true. The ‘ must ’ expresses the formal condition.
The study of formal conditions is the special business
of the logician. It is the purpose of this chapter to
make clear the nature of formal conditions.

In the preceding discussion the words ‘ true ’ and
‘ false ’ have been frequently used. It is to be
assumed that the reader knows quite well how to use
these words.1 It is sufficient for our purposes to
notice that whatever can significantly be said to be
true, or false, is a projyosition. The answer to a ques-
tion is always a proposition. Whenever a person
makes a statement he is putting forward a proposition
as true. Commands, requests, prayers, and questions
are not propositions. Of none of these could truth or

1 The determination of what ‘ true ’ and ‘ false ’ respectively
mean is a philosophical problem, which lies outside the scope of
this book.

THE IMPORTANCE OF FORM

23

falsity be significantly asserted. Using the notions of
truth and falsity we can define the relation upon
which deductive inference depends. This is the rela-
tion of implication, or — as we shall often call it —
entailing. A given proposition entails another pro-
position when there is between them such a relation
that the truth of the given proposition is inconsistent
with the falsity of the other proposition. This rela-
tion of entailing holds between the premisses, taken
together, and the conclusion in the argument about
the cutter, given on page 21. The premisses entail the
conclusion ; the conclusion follows from the premisses.
The relation of entailing is very important when we
want to make use of knowledge we already possess in
order to discover something we did not know. If we
can find a proposition which entails another, and if
we know that the entailing proposition is true, then
we know that the entailed proposition is true. Sup-
pose we know that all cutters are single-masted. We
can see at once that whatever is not single-masted is
not a cutter. Each of these propositions entails the
other. It would thus be irrational to accept one and
reject the other. In the case of this simple example
no one would be likely to do so. Indeed, the reader
may feel that to say ‘ whatever is not single-masted
is not a cutter ’ is only a more awkward way of saying
‘ all cutters are single-masted ’. Let us consider
another example : If the fire was caused by a fused
wire, then it would spread along the electric cables. This
proposition entails the proposition : If the fire did not
spread along the electric cables, then it was not caused
by a fused wire. Each of the italicized statements is
a single proposition. Either entails the other. The
fact that the one entails the other is quite independent

24

LOGIC IN PRACTICE

of its truth or falsity. We may be mistaken in sup-
posing that there is such a connexion between the
fusing of the wire and the spread of the fire along the
cables ; whether, or not, there is such a connexion
depends upon material conditions relating to the
behaviour of fused wires, fire, electric cables. The
ascertainment of these conditions requires special
knowledge concerning matters of fact. But the truth
of the statement that one of the above propositions
entails the other is quite independent of matters of
fact ; the statement is to the effect that a formal
relation holds between the two propositions whether
they are both true or whether they are both false. This
formal relation is such that we must accept both or
reject both, since either entails the other.

One proposition may entail a second although the
second does not entail the first. For example, All
sailors are superstitious entails Some sailors are super-
stitious, but not conversely. Thus the relation of
entailing is not simply reversible, although two
propositions may entail each other. As soon as we
understand the nature of entailing, we can formulate
a fundamental logical principle, namely, Whatever is
entailed by a true proposition is true. We shall call this
the Principle of Deduction, for it is in virtue of this
Principle that we can validly infer one proposition
from another. This Principle lies at the basis of all
conclusive reasoning. The reader may never have
met the Principle in this abstract form, yet he will
often have reasoned in accordance with it. We apply
the Principle whenever we argue that a given con-
clusion must be accepted because certain premisses
have been accepted. We may, of course, be mistaken
in supposing that the premisses are thus related to

THE IMPORTANCE OF FORM

25

the conclusion. Various ways in which we are prone
to make this mistake will be mentioned in the next
two chapters. Here it is sufficient to notice that this
Principle is a formal condition of deductive inference.

Let us go back to the committee of investigation
whose deliberations were sketched in the last chapter.
We have now to notice that their thinking was con-
trolled by formal, no less than by material, conditions.
No doubt they were not explicitly aware of these
formal conditions, but their thinking was effective
only in so far as it was in accordance with them. The
committee would have been simply stupid if, having
guessed that the fire was caused by a lighted match,
and having admitted that, in that case, the fire would
have broken out either in a cabin or in a public part
of the ship, they had nevertheless stuck to this guess
although it had been ascertained that the fire broke
out in the luggage-room, i.e. neither in a cabin nor
in a public part of the ship.1 The way in which their
thinking was controlled by formal conditions may be
clearly shown if the steps of their reasoning are set
out at the tedious length required to exhibit all the
conditions determining the direction of their thinking.
This we can do in the following manner :

Problem : Something, we don’t know what,
happened, and then, fire occurred. (An
observed fact.)

Question : What happened ?

First Guess : An unnoticed lighted match came
into contact with an inflammable part of the
ship, and set fire to it.

1 The reader should consider this long statement, in order to
see why, if the committee had so reasoned, they would have been
stupid.

26

LOGIC IN PRACTICE

I. Testing the Guess :

(1) If so, then the match was dropped in a cabin
or in a public part of the ship, and the fire be-
gan in the place where the match was dropped.

(2) But, the fire broke out in the luggage-room
(i.e. not in a cabin nor in a public part of the
ship).

(3) Therefore, the cause of the fire was not a
lighted match.

Second Guess : A wire fused.

II. Testing the Guess :

(1) If so, then the fire would spread along the
electric cables.

(2) But, the fire did not spread along the electric
cables.

(3) Therefore, the cause of the fire was not a
fused wire.

Third Guess : Some one deliberately set the ship
on fire.

Further Question : Who could want to set a ship on
fire ?

Tentative Answer : Some one who would benefit by
its destruction.

Further Question : Who would benefit in this case ?

III. Testing the Guess {in the light of the further

questions) :

(1) If the ship were deliberately set on fire, some
one would benefit by its destruction.

(2) If the ship were over-insured, the owners
would benefit by its destruction.

(3) But the ship is over-insured.

(4) Therefore, the owners benefit by its destruc-
tion.

THE IMPORTANCE OF FORM

27

What exactly has this reasoning established, and
how has it established it ? It must be observed that,
so far as we have gone, it has not been shown what
the cause was, nor that the owners had anything,
directly or indirectly, to do with the burning of the
ship. All that has been shown is that certain possible
causes of fire on board ship were not, given certain
assumptions, the actual cause of this particular fire.
We have now to examine the way in which the
reasoning proceeded. It should be noticed that the
committee, faced with the problem, did not imme-
diately obtain premisses which entailed the answer to
their question. On the contrary, they had to jump to
a possible conclusion, and then test its correctness.
Each successive guess led to the formulation of an
hypothesis, regarded as having certain consequences.
The reader will see that the reasoning in I and II
proceeds in precisely the same way. In each case the
hypothesis is rejected because the consequence was
found not to be the case. This reasoning may be
schematically represented, in a shortened form, if we
use Hj, H2 to stand respectively for the first two
hypotheses, and Cj, C2 for their corresponding conse-
quences. The scheme is :

(I) If Hj, then Cl5

(II) If H2, then C2,

but not Cj,
not Hj.

but not C2,
not H2.

It is easy to see that in both cases the reasoning is in
conformity with the principle, whatever is entailed by
a true proposition is true. This may be shown shortly
by using H for either Hj or H2, and C for either Cx
or C2. Provided that If H, then C is true, it follows
that C is true, for C is only part of what the original

28

LOGIC IN PRACTICE

proposition, If H, then C, asserts ; hence, if this
assertion is true, the part of it must be true also. The
Principle of Deduction tells us that we must not
accept a given proposition and reject another entailed
by it. To reject C is to say that C is false ; this is
equivalent to saying, Not C is true. For example, to
assert This paper is white is false is to assert that This
paper is not white is true. We cannot reject the
consequence and accept the hypothesis.

The reasoning in I and II is said to have the same
form. The reasoning in III is of a different form,
which will be discussed later.1 The reader will have
had no difficulty in recognizing the soundness of the
reasoning in each case. It is easier to see that an
argument is sound, or unsound, than to see wherein
its soundness, or unsoundness, consists. But to have
insight into the conditions of sound reasoning is very
important for us as practical logicians.

It must be noticed that in discussing the conditions
upon which the soundness of an argument depends
we do not need to consider a particular argument.
We need not have taken the example of the ship on
fire ; we might have discussed the authorship of the
Book of Job. Hence we could let ‘ H ’ stand for either,
or both, of H1? and H2, i.e. we were able to drop the
numerals which tied an H to a particular example.
What we said about the reasoning in I and II was
quite general ; it related to a form of reasoning. Very
many arguments could be fitted into this form. The
conclusiveness of an argument depends entirely upon
its form. Sound reasoning is valid reasoning.

The validity of reasoning depends upon purely
formal conditions. These conditions are quite general,

1 See p. 88 below.

THE IMPORTANCE OF FORM

29

and are thus formal ; hence they are independent of
special matters of fact.

All reasoning, when fully stated, has a formal
aspect. This does not mean to say that all reasoning
is deductive, although all conclusive reasoning is deduc-
tive. It means that if our reasons are sound reasons
in a given case, they must be sound in the case of any
other argument which has the same form. The
notion of the form of an argument is not familiar to
most people. It is an abstraction. A quite simple
example may show how the validity of our reasoning
depends, not upon the matter-of-fact assertions we are
prepared to make, but upon the form of the reasoning.

We will suppose that A and B are sitting on a rocky
cliff on the Cornish coast. A says, ‘ There are blasting
operations going on.’ B says, ‘ How do you know
that ? ’ A replies, ‘ Because blasting always sounds
like that.’ B says, ‘ But the sea rushing into the
clefts underneath makes a sound like that.’ A main-
tains, ‘ No. It isn’t the sea ; it is the sound of
blasting.’ B objects, ‘ Well, anyhow, you haven’t
proved your point. Even if blasting does sound like
that, so does the sea when it rushes in underneath the
cliff.’ If at this stage of the argument A makes the
counter- objection that the sound of the sea dashing
underneath isn’t exactly like the sound of blasting.
B may well reply that A’s reason was not a good
reason. It was not a good reason because A should
have said, ‘ Only blasting makes a sound like that.’

We will suppose that A is obstinate and stupid, and
that B is patient and of a pedagogical turn of mind.
A continues to maintain that his original reason was
a good reason. Whereupon the following dialogue
takes place.

30

LOGIC IN PRACTICE

A : ‘I don't see any difference between All
blasting sounds like that and All that sounds like that
is blasting, except that the second way of putting it
is very clumsy.’

B : ‘ Do you see any difference between All seals
are mammals ’ and ‘ Only seals are mammals ? ’

A : ‘Of course. The first is true, and the second
isn’t.’

B : ‘ Why do you say the second proposition isn’t
true ? ’

A : ‘ Because men, and horses, and elephants, and
a lot of other animals are mammals as well as seals.’

B : ‘ Then it doesn’t follow from All seals are
mammals that All mammals are seals ? ’ 1

A : ‘Of course not.’

B : ‘ Then you ought to admit that it doesn’t follow
from the fact that all blasting operations make a certain
sort of sound, that whatever makes that certain sort of
sound is a blasting operation .’

It is to be hoped that B’s argument may have
convinced the reader, whatever may have been the
case with A. The point that concerns us is why B
began to talk about seals and mammals in order to
show A that his reason for holding that a certain
sound was due to blasting was not a good reason.
There is no connexion between blasting and sounds,
on the one hand, and seals and mammals, on the other.
B's purpose was to call A’s attention to the form of
what he said, since, if the reason he offered was a
good reason, it must be a good reason in any other
argument of the same form. Now all seals are mammals

1 Here B assumes that if what follows from a given proposition
is false, then that given proposition is also false, and A’s next
remark accepts this assumption.

TIIE IMPORTANCE OF FORM

31

is related to all mammals are seals in the same way as
all blasting operations make that sort of sound to
whatever makes that sort of sound is a blasting operation.
We can abbreviate the statement if we substitute S
for seals, and M for mammals. Then we can see that
all S is M does not entail all M is S, nor conversely.
In no case does all M is S follow from all S is M, no
matter what S, or M, represents, and no matter,
therefore, whether it is in fact the case both that
all S is M and that all M is S.

We are able to say ‘ in no case does it follow ’,
because whether it does follow or not depends, not
upon the matter of fact asserted, but wholly upon
the form of the assertion. It is for this reason that
the logician must insist upon the importance of form,
since his purpose is to determine the validity of
reasoned arguments. The conditions of validity
constitute the formal conditions of a problem.

CHAPTER III
DEDUCTIVE FORMS

* All the inventions that the world contains,

Were not by reason first found out, nor brains ;

But pass for theirs who had the luck to light
Upon them by mistake or oversight.’

Samuel Butler (1612-80)

THE objection to giving bad reasons is not to be
found in the falsity of the conclusion. On the
contrary, sometimes bad reasons are given in order
to support a conclusion which is in fact true. The
objection is that bad reasons do not show that the
conclusion is true. ‘ Bad reasons ’ are not properly
reasons at all, since their badness consists in their not
affording any reason why we should accept the con-
clusion. Accordingly, if we are shown the unsound-
ness of the argument, we shall be left without any
ground for the acceptance of the conclusion. Of
course if we firmly believe that the conclusion is true,
we may then look round for reasons to support it. In
this search we may be helped if we know the kind of
premisses which are required to justify the acceptance
of the conclusion. The kind of premisses required
will depend upon the form of the argument into which
the premiss has to be fitted.

Reasoning is possible because the truth, or falsity,
of one proposition is not independent of the truth, or
falsity, of all other propositions. Every statement
we make has consequences, i.e. implies that other
statements are true and still others false. Most state-
ments we make have grounds, i.e. are related to other

32

DEDUCTIVE FORMS

33

statements which imply them. We often are not
aware of these grounds, nor of these consequences.
In reasoning, however, we seek grounds or we seek
consequences. This should be clear from our dis-
cussion of inference in Chapters I and II.

There are seven possible relations which may hold
between any two propositions with regard to the
inferability of one from the other. Every one is
familiar with these relations, even if he happens
not to know the technical names which logicians
have used for the sake of distinguishing between
them.

We will begin by considering the two opposite
relations of compatibility and incompatibility. Two
characteristics are incompatible when the presence of
one necessitates the absence of the other, and con-
versely. The following statement illustrates a com-
mon use of the word : ‘ He felt that to be a politician
and a preacher of righteousness was to combine two
vocations practically incompatible.’1 If the reader
thinks that a man may be both a politician and a
preacher, he thinks that these characteristics are
compatible. Compatible and incompatible have the
same significance when asserted of propositions. One
proposition is incompatible with another if they cannot
be true together. Propositions may, however, be
compatible without being so related that it is possible
to infer the one from the other, or to infer from the
truth, or falsity, of the one to the truth, or falsity, of
the other. The relation of bare compatibility interests
no one except a logician. The proposition Darwin
wrote an important book is compatible with the pro-
position The traffic problem in New York is insoluble,
1 Taken from the Shorter O.E.D.

3

84

LOGIC IN PRACTICE

and both of these with the proposition Some school-
children like to study logic. But this bare compati-
bility is uninteresting, because nothing else follows
from it. That is why the disconnected remarks of a
Mrs. Nickleby or a Miss Bates are apt to be boring,
and why some old gentlemen’s stories are pointless.
The relation of bare compatibility cannot afford a
basis for inference, nor provide the material for a joke.
Nevertheless, it is necessary to notice that such a
relation does hold between certain propositions. Two
propositions thus related are said by logicians to be
independent.

The most extreme degree of compatibility holds
between propositions which are equivalent. Examples
are : All poets are sensitive to criticism, No poets are
insensitive to criticism ; If Roosevelt abolishes war
debts, the Americans will be displeased. Either the
Americans will be displeased or Roosevelt will not
abolish war debts.1 An examination of these two
examples will show that, in each case, the truth of
the second proposition follows from the truth of the
first, and converse^ ; the falsity of the second
proposition follows from the falsity of the first, and
conversely. Hence, to assert either entails the asser-
tion of the other. Thus two propositions are equiva-
lent if one entails the other, and conversely.

Two propositions are compatible if one can be
inferred from the other, even though the other cannot
be inferred from the former. Thus, Some poets are
sensitive to criticism can be inferred from All poets are
sensitive to criticism. The reverse inference is not,
however, permissible. Hence, the relation of the first

1 Examples of equivalent propositions will be found on pages
45-46 above.

DEDUCTIVE FORMS

35

of these two propositions to the second is different
from the relation of the second to the first. The two
relations must, then, be distinguished. One proposi-
tion is sub-implicant to a second if the first can be
inferred from the second, but not conversely. In this
case, the second is super -implicant to the first. The
relations are called respectively siib -implication and
super -implication.

Two propositions may be compatible although
neither can be inferred from the other, and yet they
are not independent. This is so when the two
propositions are so related that they cannot both be
false and may both be true. For example, some
stupid people are obstinate and some stupid people are
not obstinate. If we know that one of these proposi-
tions is false, we can infer that the other is true ; but
if we know that one is true, we cannot infer that the
other is true, nor that it is false. Both possibilities
remain open. Fortunately this is plain common
sense. Logicians have unfortunately invented the
inappropriate word ‘ sub-contrary ’ to express this
relation.

Incompatible propositions cannot be inferred from
each other. Since, however, of two incompatible
propositions one at least must be false, knowledge
that one of them is true enables us to infer that the
other is false. We must distinguish between incom-
patible propositions which are contrary and those which
are contradictory. Two incompatible propositions are
contrary if neither need be true and both cannot be
true. Examples are : No economic theories are sound,
All economic theories are sound ; Tobias Fortescue
always tells lies, Tobias Fortescue never tells lies.
To each of these pairs of propositions there is

36

LOGIC IN PRACTICE

obviously an alternative. Possible Tobias Fortescue
sometimes tells lies and sometimes speaks the truth.
Two incompatible propositions are contradictory if
one must be true and one must be false. It follows
that from knowledge of the truth, or falsity, of the
one the falsity, or truth, of its contradictory can be
inferred. Examples are : Whoever trusts a 'poli-
tician's promises shows himself to be foolish, One may
trust a politician's promises without showing oneself to
be foolish ; St. Paul's Cathedral is smaller than St.
Peter's at Rome, St. Paul's Cathedral is either the same
size as, or is larger than, St. Peter's at Rome. A
proposition is denied when either its contradictory or
a contrary to it is asserted.

The seven relations, with regard to inferability,
which may hold between two propositions are, then :
(1) independence, (2) equivalence, (3) sub-implication,
(4) super-implication, (5) sub-contrariety, (6) con-
trariety, (7) contradiction. When (1) holds, no
inference is possible ; when (2), (3), or (4) holds, at
least one of the two propositions implies the other ;
when (5), (6), or (7) holds, neither proposition implies
the other, but — under the various conditions specified
above — it is possible to infer something with regard
to the falsity, or with regard to the truth of one
proposition given knowledge of falsity or truth
concerning the other proposition.

We have next to consider affirmation and denial.
These notions are familiar to every one. Any ques-
tion theoretically admits of being answered by a
‘ Yes ’ or by a ‘ No ’, although sometimes we may be
in doubt as to which answer is correct.1 The Yes-
answer is in effect an affirmation ; the Ao-answer is

1 See p. 96 seq.

DEDUCTIVE FORMS

37

in effect a denial. For example, the reader may be
asked, ‘ Is it worth while to study logic ? ’ If he
answers, ‘ No ’, then he is in effect saying that it is not
worth while to study logic ; if he answers, ‘ Yes ’,
then he is in effect saying that it is worth while to
study logic. It is what the question is about that
determines whether the answer is an affirmation or
a denial. The question, cited as an example, is about
the worth-whileness of studying logic. The questioner
wants to know whether the property of being worth
while belongs, or not, to the study of logic. A denial
that the property belongs might be expressed by the
sentence, ‘ It is a waste of time to study logic ’, or
by the sentence, ‘ Studying logic is unworth while ’.
These sentences are affirmative, but the speaker who
uses one of them to express his answer to the given
question is making a denial. The distinction between
affirmative and negative sentences derives its signifi-
cance from the distinction between affirmation and
denial. In affirming, or denying, we use sentences,
but what we affirm, or deny, is not a sentence but
what the sentence is used to express. In answering a
question the sentence is used to express a proposition.
A proposition expressed by an affirmative sentence is
usually called an affirmative 'proposition ; one ex-
pressed by a negative sentence is called a negative
proposition. In dealing with propositions out of the
context in which they may be asserted, this is a
convenient procedure. But it must not mislead us
into supposing that the same state of affairs cannot be
referred to both by a negative and by an affirmative
proposition. On the contrary, every affirmative state-
ment can be translated into a corresponding negative
statement (and conversely), which is equivalent

38

LOGIC IN PRACTICE

to the original. For example : Question — ‘ Are
philosophers consistent ? ’ Answer — ‘ No. Some
philosophers are inconsistent.’ The answer might
have taken the form, ‘ Some philosophers are not con-
sistent In the first form, being inconsistent is
affirmed of some philosop>hers ; in the second form,
being consistent is denied of some philosophers. Both
statements refer to the same characteristic of these
philosophers. The negative statement must deny the
possession of the opposite property to that which the
affirmative statement asserts to be possessed. Such
an opposite property is called a contradictory property,
e.g. consistent and inconsistent are contradictory. But
not consistent is equivalent to inconsistent. Which of
the two contradictory properties we shall assert of a
subject depends upon the question determining our
thinking, but whether the assertion is to be an
affirmation or a denial will depend upon the matter of
fact to be asserted.

It follows, then, that we have not denied the
fundamental distinction between affirmation and
denial. On the contrary, we have insisted upon it in
maintaining that to affirm any property of something
is equivalent to denying the possession of the contra-
dictory property. Two propositions thus related are
said to be obverses of each other ; the process of draw-
ing one of these propositions from its equivalent is called
obversion. An inference from a single proposition to an-
other implied by it is called an immediate inference.
The name is not fortunate ; it may, however, serve to
remind us that in asserting any proposition whatever we
are committed to other assertions, namely, to whatever
is implied by the original proposition whether we happen
to have noticed these implications or not.

DEDUCTIVE FORMS

89

Statements about things of a certain sort are, we
have seen, statements about classes, e.g. philosophers.
With regard to any property which we could signifi-
cantly think of as belonging, or not belonging, to
philosophers, there are three possibilities. Every
philosopher might possess the property ; or none
might ; or some might possess it and some not. For
example, Some philosophers are hot-tempered and
some are not ; No philosophers are consistent ; All
philosophers are liable to headaches. In denying that
every philosopher possesses a given property, we
commit ourselves to the assertion that some do not
possess it ; in denying that some possess it, we affirm
that none do. In short, to deny a given statement is
to affirm its contradictory.

In the preceding discussion we have been making
use of two fundamental logical principles, which
together determine the nature of contradiction. These
principles have received technical names. They may
be stated as follows :

I. Principle of Non-Contradiction : Given any

proposition, P, then P cannot be both true and
false.

II. Principle of Excluded Middle : Given any pro-
position, P, then P is either true or false. A
third alternative is excluded, since there is no
mean between truth and falsity.

With regard to a given proposition we may not
know whether it is true or whether it is false ; we
know, however, that it must be one or the other and
that it cannot be both.

These principles may be stated in a form which is
directly applicable to the possession bv an object of

40

LOGIC IN PRACTICE

a property. Let 0 be any object, and F any property
which could be significantly asserted of 0.1 It wall be
convenient to represent the property contradictory
to F by ‘ non-F Then we get :

I 0 cannot possess both F and non-F.

II 0 must possess either F or non-F.

The class philosophers was taken as an example,
just as in the preceding chapter we took psycho-
analysts, seals, mammals, blasting operations, ships.
We could have taken any other class. The character-
istics distinguishing one sort of thing from another
sort have throughout been irrelevant to our discus-
sion, since we have been concerned only with the form
of the assertion. What we have said applies quite
generally to any class and to any property. We can
show that our statement is quite general by using
symbols to stand for any class, just as in algebra we
use symbols to stand for any number. It was in this
way that we used symbols in formulating the two
principles of non- contradiction and excluded middle.

Using X and Y to stand for any two different
classes, we can use All X’s are Y’s to represent All
philosophers are thinkers, or All civil servants are
patriotic, or any other statement of the same form. In
fact, what All X's are Y’s represents just is the form,
which is common to ever so many different statements,
namely, to those which assert that every member of
one class is included in some other class. It is this

1 It is not significant, i.e. does not make sense, to assert some
properties of some objects. For example, it is nonsensical to say,

‘ Some courageous acts are triangular and equally nonsensical
to say ‘ Some courageous acts are non-triangular ’. The property
of being triangular could be significantly affirmed, or denied, only
of that which has shape.

DEDUCTIVE FORMS 41

sameness of form which alone is relevant to the
purposes of a logician. Similarly we can represent by
Some X’s are Y’s what is common to all statements,
to the effect that some members of a given class are
included in some other class.

A proposition of the form All X’s are Y’s is called
a universal proposition ; one of the form Some X’s
are Y’s is called a particular proposition. The dis-
tinction between universal and particular propositions
is very important. Upon it depends the fact that an
inductive generalization from some members of a class
to all is an inference going beyond the evidence. It
is not, however, the case that in asserting that some
X’s are Y’s we always intend to assert that only some
are. If it were, then, we could never pass from the
assertion that Some psychoanalysts lack humour to the
assertion that all do. Although the evidence may be
sufficient only to justify the assertion about some,
nevertheless the universal assertion might be in fact
true. Thus the assertion of some is not necessarily
incompatible with the assertion of all. The force of
‘ some ’ is selective ; its use enables us to make a
partial generalization, leaving open the possibility
that a universal generalization could also be truly
asserted. In using symbolic expressions it is im-
portant that each symbol should have a fixed refer-
ence. Accordingly, we have to decide whether to
interpret ‘ some ’ in the sense ‘ some, it may be all ’,
or in the sense ‘ some, but not all For the reasons
given, it is convenient to adopt the former interpreta-
tion. Then, if we wish to say that some but not all
X’s are Y’s, we assert the compound proposition.
Some X’s are Y’s and some X’s are not Y’s. It is,
indeed, easy to see that Only some socialists are

42

LOGIC IN PRACTICE

Marxians denies both that all socialists are Marxians
and also that no socialists are Marxians. The sen-
tence is grammatically simple, but the proposition
expressed is compound.

These various statements about classes can be
symbolically represented as follows :

(1) All X'sareY's.

(2) No X's are Y's.

(3) Some X’s are Y's.

(4) Some X's are not Y's.

We may briefly summarize what has already been
said about these four forms. (1) and (2) are forms of
universal, or unrestricted, generalizations. (3) and
(4) are forms of partial, or restricted, generalizations.
(1) and (3) are affirmative ; (2) and (4) are negative.
Negative propositions of these forms can be regarded
as denying inclusion, i.e. as asserting exclusion of one
class from another. Looked at from this point of
view, we can see that the four forms are derived from
the fact that, with regard to any class X, we can
assert that X is either wholly or partially included in,
or excluded from, the class Y.

Any one of these four propositions can be regarded
as consisting of two terms combined in the way
appropriate to its special form. For example, No
civil servants are members of Parliament may be said
to be about civil servants and members of Parliament.
Since what is said about them is that the one class is
wholly excluded from the other, it is clear that No
members of Parliament are civil servants is equivalent
to the original proposition. In the context of a dis-
cussion, which of the two propositions we chose to
assert would depend upon the question determining

DEDUCTIVE FORMS

48

the direction of our thinking. If we were asking :
‘ Are any civil servants members of Parliament ? ’
we should probably assert the original proposition ;
if we were asking ‘ Are any members of Parliament
civil servants we should assert the second. The
term which comes first is often called the subject , the
second term the 'predicate . In this usage of the words
‘ subject ’ and ‘ predicate ’ we think of the proposition
as asserting something about something. That which
is asserted about something is the predicate ; that
about which something is asserted is the subject.1

It should be observed that we have used ‘ about ’
in two different senses. In the former sense the
proposition is ‘ about ’ both the terms ; in the latter it
is ‘ about ’ the subject-term. The narrower sense is
due to the fact that the question which determines
our thinking is a question concerning the subject-
term. In the context of a discussion the subject-term
is not always stated first ; which of the terms is
subject must be decided by reference to a question
which the given proposition might be regarded as
answering. In considering propositions, taken in
isolation for the purposes of example, we shall assume
that the subject-term is the first term.

We have seen that in the case of the proposition
No civil servants are members of Parliament, the
subject- and predicate-terms can be interchanged
without alteration in the truth, or falsity, of the
proposition asserted. In general, any proposition of
the form No X’s are Y’s is equivalent to No Y’s are
X’s. These propositions are said to be simple con-
verses of each other. Their equivalence is due to the
fact (i) that exclusion is a reversible — or, as logicians
1 Here ‘ predicate ’ is used in a wide sense.

44

LOGIC IN PRACTICE

say, a symmetrical — relation, so that in excluding one
class from a second, the second is ipso facto excluded
from the first ; (ii) that every member of the class
indicated by the subject-term is asserted to be
excluded from the class indicated by the predicate-
term. Where the reference is to every member of the
class, the term is said to be distributed.

Inclusion is not a symmetrical relation. Accord-
ingly, the predicate of an affirmative proposition
(which asserts inclusion in a class) does not refer
distributively to every member of the class indicated
by the predicate-term. In the particular affirmative
proposition Some X’s are Y’s, the prefix some , how-
ever, restricts the reference of the subject-term just
as the reference of the predicate-term is restricted.
Accordingly, this proposition also admits of simple
conversion. Some X’s are Y’s is thus equivalent to
Some Y’s are X’s. Since the prefix all indicates un-
restricted reference, the subject of a universal affirma-
tive proposition is distributed. Being affirmative,
however, the predicate of All X’s are Y’s is not distri-
buted ; hence these terms are not simply inter-
changeable. It follows that this proposition has no
simple converse.

For a similar reason the particular negative proposi-
tion Some X’s are not Y’s is not simply convertible.
The negative form, expressing exclusion, does involve
reference to every Y ; the prefix some restricts the
reference of the subject-term, X, which cannot,
therefore, become the predicate of a negative proposi-
tion. For example, the assertion that some philoso-
phers are not rich is consistent with there being some
who are rich and also with there being some who are
rich without being philosophers. Since the original

DEDUCTIVE FORMS

45

assertion leaves open both these possibilities, to
interchange the terms would be to go beyond the
evidence.

It should be clear that whether a proposition is
simply convertible, or not, depends upon the distribu-
tion of the terms. If the reference of both terms, or of
neither term, is distributive, then the proposition is
simply convertible. But if one term is distributed,
whilst the other is not, then the proposition has no
simple converse. In this case, however, the proposi-
tion has a negated simple converse equivalent. For
example, All polite people are tactful is equivalent to
No untactful people are polite. In totally including
those who are polite in the class of those who are
tactful, we ipso facto exclude the untactful from the
polite. Likewise, Some great statesmen are not free from
vanity is equivalent to Some who are not free from
vanity are great statesmen. If the reader looks back to
what was said about ob version, he will see that these
equivalents are converted obverts of each other.1

The converse equivalents may be summed up in the
following schema :2

Simple Converse Equivalents.

No X’s are Y’s = No Y’s are X’s.

Some X’s are Y’s = Some Y’s are X’s.

1 A converted obvert is sometimes called a contrapositive.
Since every proposition can be obverted, the contrapositive can
be again obverted. The obverted contrapositive of All polite
people are tactful is All untactful people are impolite. The reader
will easily be able to derive other examples. For a fuller dis-
cussion, see L. S. Stebbing : A Modern Introduction to Logic,
Chap. V, § 2.

2 In this schema, the sign == is used as a shorthand symbol
for 1 is equivalent to ’. It will often be found convenient to use
this abbreviation.

40

LOGIC IN PRACTICE

Negated Converse Equivalents.

All X’s are Y’s = No non-Y’s are X’s.

Some X’s are not Y’s = Some non-Y’s are X’s.

The reader should observe that the equivalence of
the propositions, in each of these pairs, results from
the distributive reference of their respective terms.
In deductive inferences distribution is of funda-
mental importance, since to infer a proposition, con-
taining a distributed term, from a premiss in which
that term was given undistributed would be to go
beyond the evidence. The facts concerning distribu-
tion, in the case of each of the four forms of proposi-
tions given above, may be summarized as follows :
(1) The •predicate-term, is distributed in a negative, but
undistributed in an affirmative proposition ; (2) the
subject-term is distributed in a universal, but undis-
tributed in a particular proposition.

Bearing these points in mind, we may ask what
exactly is the information given us by the statement
that All polite people are tactful. It informs us that :

(1) If anyone is polite, he is tactful.1

(2) If anyone is not tactful, he is not polite.

It leaves open the two possibilities :

(i) That someone is tactful although not polite.

(ii) That no one is tactful without being polite.2

1 This is just another way of saying * All polite people are
tactful whilst (2) is another way of saying ‘ All untactful people
are unpolite ’. Hence, (1) and (2) are contrapositivea of each
other, and are thus equivalent.

a The two possibilities are left open because the predicate-term
of the original statement is not distributed. Accordingly, whilst
the polite are restricted to the tactful, the converse is not the case.
The reader will find it worth while to convince himself of the

DEDUCTIVE FORMS

47

Not both these possibilities can be realized, but the
given statement about polite people does not tell us
which is in fact the case. It is clear, however, that, if
we know with regard to a certain person, say Ramsay
MacDonald, that he is polite, we can deduce that he
is tactful. Again, should we happen to know with
regard to another person, say Tobias Fortescue, that
he is not tactful, then we could deduce that he is not
polite. But if we knew only that he is not polite, we
could not (on the sole basis of the generalization about
polite people) deduce that he is tactful, nor that he
is not. That this deduction would be invalid follows
from the fact that possibility (i) was not excluded by
the original statement.

The arguments suggested above are examples of a
very common form of argument, which has been
technically called syllogism. A syllogism may be re-
garded as essentially consisting in the application of a
generalization (or a general rule) to a specified case
in order to deduce a result. We may begin by con-
sidering an example :

(Rule) All aviators are intrepid.

(Case) Amy is an aviator.

(Result) Amy is intrepid.

Suppose we deny that Amy is intrepid. Then we
must, in consistency, deny either that she is an aviator
or that all aviators are intrepid. Then we get :

(Denial of Result) Amy is not intrepid.

(Case) Amy is an aviator.

(Denial of Rule) Some aviators are not intrepid

truth of these contentions, and to see that their truth is a conse-
quence of what was said in the discussion of obversion and the
converse equivalents.

48

LOGIC IN PRACTICE

provided that we keep to the assertion of the Case.
If, however, we are prepared to maintain that all
aviators are intrepid, we must accept the following
argument :

(Rule) All aviators are intrepid.

(Denial of Result) Amy is not intrepid.

(Denial of Case) Amy is not an aviator.

The reader will have no difficulty in seeing that each
of these three arguments is valid, i.e. the acceptance
of the premisses entails the acceptance of the con-
clusion. Accordingly, the denial of the conclusion
entails the denial of at least one of the premisses. Thus,
denial of the Result, combined with acceptance of the
specified Case, entails denial of the Rule ; denial of
the Result, combined with acceptance of the Rule,
entails denial of the specified Case. This point may
be put in a different, but equivalent way. The three
propositions : (1) All aviators are intrepid, (2) Amy
is an aviator, (3) Amy is not intrepid, cannot be true
together. The combination of any two of them entails a
conclusion which contradicts the omitted proposition.

The principle in accordance with which the first of
these three arguments proceeds may be formulated as
follows : Whatever can he asserted (affirmatively or
negatively) of any member of a given class can he like-
wise asserted of any specified member. This is called
the Applicative Principle, since it permits us to apply
to a specified case whatever is asserted of every
case in general.1 The Principle yields the symbolic
form :

1 This Principle is also called the Principle of Substitution, for
it is the Principle in accordance with which values can be substi-

tuted for variables in, for example, ‘(a-fb) (a — b) = (a2 — b2) ’.

DEDUCTIVE FORMS

49

If Anything which is a member of X has F (or not),
and A is a member of X ;
then A has F (or not).1

The bracketed ‘ or not ’ shows that the form is valid
whether the property be affirmed or denied of the
members of X, but that it must be in like manner
affirmed or denied of A.

An allied principle — which may be called the
Principle of Excluding an Individual from a Class —
covers the case of deducing that Tobias Fortescue is
not polite since he is not tactful, and all polite people
are tactful. The Principle may be formulated as
follows : If a given individual lacks (or possesses) a
property which any member of a certain class possesses
(or lacks), then that individual is not a member of that
class. This yields the form :

If Anything which is a member of X has F (or not),

and A has not F (or has) ;
then A is not a member of X.

We may now consider another type of syllogism, in
which the specified case may be regarded as replaced
by a set of cases of the same sort, this set being a class
falling within a wider class. For example :

All intrepid people are admirable.

All aviators are intrepid.

.'.All aviators are admirable.

No civil servants are M.P.s.

Some who direct the Government are civil
servants.

.'.Some who direct the Government are not M.P.s.

1 Here (and subsequently) X stands for any class, F for any
property, A for any specified, individual.

4

50

LOGIC IN PRACTICE

The principle in accordance with which this reasoning
proceeds may be formulated as follows : Whatever can
be asserted of every member of a class can in like manner
be asserted of every sub-class contained in that class.
This principle has been named the Dictum de omni
et nullo. It yields the form :

If Every Y is Z (or not),

and Every (or some) X is Y ;

then Every (or some) X is Z (or not).

An examination of this form shows that a connexion
is established between X and Z on the ground of the
connexion between Y and Z, on the one hand, and
between X and Y, on the other. Accordingly, Y,
which occurs in both premisses but not in the con-
clusion, is called the middle term ; X and Z are called
the extreme terms. Unless the middle term is distri-
buted, no connexion is secured ; at least one of the
extreme terms must be given as having a relation to
an unrestricted reference of Y. An example may
make this clear :

Some intelligent people are witty.

All civil servants are intelligent.

.'.All civil servants are witty.

In this argument the conclusion does not follow. The
intelligent witty people may not include any of the
civil servants ; it might even be the case (so far as
the evidence provided by the premisses goes) that the
civil service, whilst requiring intelligence, deadens wit.
The conclusion is consistent with the premisses, but
not entailed by them. This may be exhibited by a
diagram in which intelligent people are collected into
one circle, witty people into another. The first

DEDUCTIVE FORMS

51

premiss ensures that these circles must at least overlap.
Thus we get :

The state of affairs asserted by the second premiss is
consistent with the three cases : (i) all civil servants
are in the overlapping portion ; (ii) none are ; (iii) some
are and some are not. Thus these premisses do not
suffice to tell us which of the three cases is correct.
If, however, intelligence were asserted to be a sufficient
condition of wit, the first premiss would become All
intelligent people are witty, and the arrangement of the
circles would become

With the emended premiss the syllogism is of the form,

Every Y is Z.
Every X is Y.
'.Every X is Z.1

This syllogism is well known to logicians under the
proper name Barbara. In order to establish a
1 In some cases the circles might coincide.

52

LOGIC IN PRACTICE

universal affirmative conclusion both premisses must
be universal affirmatives, and the terms must be
arranged as in the above schema. If the first premiss
is negative, the conclusion must also be negative.
The second premiss, being the statement that a certain
sub-class is contained in a wider class, is affirmative.
If this premiss makes an assertion about some mem-
bers of this sub-class, X, then the conclusion must
also be particular.

The arrangement of the terms in the form yielded
by the Dictum de omni et nullo is the standard arrange-
ment. It will be observed that the middle term is
predicate in the second premiss, and subject in the
first premiss. Valid syllogisms need not conform to
this arrangement. For example :

No good citizens are selfish.

People who leave fitter are selfish.

.‘.People who leave fitter are not good citizens.

If for the first premiss we were to substitute its
converse equivalent, this syllogism would be in stan-
dard form. By using the various equivalents, such
rearrangement of the position of the terms is always
possible, but it is not necessary. Other Dicta, directly
applying to other arrangements, can be formulated.1
We shall not state these Dicta here. The reader will
be able to formulate a Dictum apptying to the case
where the middle term is predicate in both premisses,
if he considers the Principle of Excluding an Indivi-
dual from a Class, which was given above. It is
sufficient here to state briefly two Rules which

1 For the statement of these Dicta, and a full discussion of
other arrangements of terms, see L. S. Stubbing : A Modern
Introduction to Logic, Chap. VI.

DEDUCTIVE FORMS

53

guarantee the validity of any syllogism. These are :

(1) The middle term must be distributed in at least
one of the premisses ; if an extreme term is distributed
in the conclusion, it must be distributed in its premiss.

(2) At least one premiss must be affirmative, and if
either premiss asserts a denial, the conclusion must
be a denial. These Rules will apply both to class
syllogisms, and also to those syllogisms in which one
term is an individual, notwithstanding the difference
in the logical properties of the two types of syllogism.1

The validity of the class syllogism depends upon
the nature of the relation of inclusion. If a plane
figure (e.g. a circle, or square, or triangle, etc.) is
wholly included in a second which is wholly included
in a third, then the first is wholly included in the
third. A relation having such a property is called
transitive. This property of transitiveness is extremely
important, for upon it depends the validity of all
deductive arguments, except those which depend upon
the Applicative Principle. The property may be
described as follows : A relation is transitive if it is
such that, given that the relation relates X and Y,
and also relates Y and Z, then it follows that X and Z
are also related by this relation. Familiar examples
of transitive relations, in addition to inclusion in, are
exclusion from, equals, greater than, exactly contem-
porary with, older than, more pious than. If a set of
policemen are arranged in order of height, beginning
with the shortest and ending with the tallest, then
(on the assumption that no two were exactly the same
height) we could pass down the row, picking out any

1 It is not possible to deal properly with these differences here.
The reader will find a discussion of the point in L. S. Stebbing :
op. cit., p. 97.

54

LOGIC IN PRACTICE

two of w'hich it would be true to say that the one
nearer the end was taller than the one nearer the
beginning of the row. This property of transitivity
yields a principle— called by William James ‘ the
axiom of skipped intermediaries He says, ‘ Sym-
bolically we might write itasa<6<c<d . . . and
say, that any number of intermediaries may be
expunged without obliging us to alter anything in
what remains written.’ 1

Implies is a transitive relation. It is in virtue of
this property of implication that it is possible to
combine a set of propositions, of which the first
implies the second, the second implies the third, and
so on, into a system such as the Euclidean system of
geometry. Many of our ordinary arguments are
intended to consist of a set of propositions wrhich
successively imply others. Usually we skip the inter-
mediate steps, and may sometimes be led to asserting
that a proposition implies another, although, had we
made the steps explicit, we should have seen that the
chain of implications had been broken.

We have so far considered deductive arguments in
which (i) one term was an individual ;2 (ii) the three
terms were all class terms. We have now to consider
arguments in which all the terms are individuals.
A proposition of which an individual is the subject is
a singular proposition, e.g. Hitler is aggressive, Hitler
admires Mussolini. The first of these is a subject-
predicate proposition ; the second is known as a

1 Principles of Psychology, Vol. II, p. 646. Cf. L. S. Stebbing :
op. cit., p. 173.

2 In this book ‘ an individual ’ means exactly what the reader
will take it to mean. In this sense a class is clearly not an indi-
vidual, but a collection of individuals, or a collection of sub-
collections.

DEDUCTIVE FORMS

55

relational 'proposition. A relational proposition is one
in which two, or more, terms are stated to be related
by some definite relation. A relational argument
consists of relational propositions, e.g. :

(i) Galileo preceded Newton.

Newton preceded Einstein.

.'.Galileo preceded Einstein.

(ii) My sister runs as fast as Tom.

Tom runs as fast as your brother.

.'.My sister runs as fast as your brother.

(iii) A is father of B.

B is father of C.

.'.A is father of C.

The reader will observe that (i) and (ii) are valid,
whilst in (iii)- the conclusion does not follow. The
premisses here imply the conclusion A is grandfather
of C. The relation father of is not transitive ; on the
contrary, it is intransitive, i.e. it is such that if it holds
between A and B, and between B and C, then it
cannot hold between A and C. Similarly, whilst older
than is transitive, older by one year than is intransitive.
From A is older by one year than B and B is older
by one year than C, there follows the conclusion A is
older by two years than C. We see this as soon as wre
understand what ‘ older than ’ and ‘ older by one year
than ’ mean. A knowledge of the system of relations
which render such arguments valid constitutes part
of our common stock of information. The plain man
is not likely to make serious mistakes simply because
of a failure to know whether a relation is, or is not,
transitive. If he does, logic cannot help him. Are
the friends of our friends also our friends ? It is not

56

LOGIC IN PRACTICE

for the logician to decide whether friendship is a
transitive relation.1

There remains a set of deductive forms, of common
occurrence in ordinary discourse, in which one of the
premisses is a compound proposition. A compound
proposition is a combination of two, or more, proposi-
tions each of which is separately assertable. A simple
proposition is one which is not compound.2 It will
be sufficient for our purpose to consider four modes
of combining simple propositions. The simplest
combination is that effected by the logical conjunction
and. The proposition Drake played bowls and subse-
quently he fought in the Armada consists of two simple
propositions ; it is true if both these propositions are
true ; false, if either of them is false. The constituent
propositions are logically independent ; the conjunc-
tive proposition merely asserts that both are true.
The other three modes of combination are effected by
the three combining forms, If . . ., then . . . ; Either
... or ; Not both. . . .

A proposition such as If Hitler defies Europe, the
League ought to intervene, is called Hypothetical. It
asserts that a certain condition ( Hitler defies Europe)
has a certain consequence ( the League ought to inter-
vene). The condition is called the Antecedent ; the con-
sequence is called the Consequent. The reader should
observe that the hypothetical proposition may be true
although neither of its constituents is true ; it is false if
the antecedent is true although the consequent is false.

1 The reader interested in the logical treatment of relations
should consult L. S. Stebbing : op. cit., Chap. X.

2 We have treated propositions of the form All X is Y, etc.,
as simple. There are objections to this procedure, but the proper
treatment of this topic would take us too far afield. See L. S.
Stebbing : op. cit., Chap. IV.

DEDUCTIVE FORMS

57

A proposition such as Either you are an optimist or
I am misinformed as to the facts is called Alternative.
This name is self-explanatory. An alternative pro-
position is true if at least one of the alternatives is
the case ; it is false if neither is the case. It is a
mistake to interpret the ‘ either . . . or ’ as exclusive.
In other words, ‘ or ’ does not exclude ‘ both ’. He is
either clever or hardworking does not exclude the
possibility that he is both ; it only excludes the
possibility that he is neither.

A proposition such as It is not the case both that a
gale is blowing and that it is safe to swim in the bay,
is false provided that each of the disjointed proposi-
tions is true, for this compound proposition asserts
that at least one of its constituents is false. It is
accordingly called a disjunctive proposition.

The conjunctive form does not give rise to any
inferences which are not redundant. Each of the
other three forms can afford a premiss in an inference,
as the reader may have observed from the statement
of its significance. This may best be shown by means
of an example :

‘ If Roosevelt’s Recovery Campaign succeeds,
then the economic system of America will be a
form of “ controlled capitalism ”. In that case both
industry will be co-ordinated and unified and also
Labour will be in a new relation to employers. If
it succeeds, then either Great Britain must follow
the example of America, or else she will become
bankrupt. The latter alternative is unthinkable.
Hence we may conclude that Great Britain must
adopt the American plan.’

The reader should have no difficulty in assigning each

58

LOGIC IN PRACTICE

of the single propositions in the above argument to
its appropriate form. The argument is not fully
stated. The passage is argumentative because certain
of the propositions are said (explicitly or implicitly)
to imply others. But the implied propositions are not
formally set out. Hence, in the strict sense this
passage does not present a formal argument ; it
presents the material for one. This is the mode of
procedure adopted in ordinary discourse and in most
argumentative discussion. It is not formal reasoning,
but it has, for the instructed hearer, the force of
formal reasoning. Whenever we use a ‘ therefore ’,
* hence ’, ‘ consequently ’, ‘ we conclude that ’, ‘ it
follows from ’, ‘ since ’, ‘ because ’, we are stating an
argument, the validity of which depends upon its
being an example of a form of implication. We seldom
state fully the premisses upon which the validity of
our argument depends. To do so wrould be tedious
and is often unnecessary. The omission, however,
sometimes leads us into drawing erroneous conclusions,
as we shall see later.1

To return to the example. The first statement does
not assert that Roosevelt’s Campaign will succeed ;
it asserts a consequence of its success, should it
succeed. The second statement asserts a conjunc-
tion of two other consequences of its success. The
third statement asserts a further consequence, in
the form of an alternation ; it does not assert which
alternative would be realized, but merely that at least
one will be. The next statement cuts out one alter-
native. The last statement explicitly draws the
conclusion which follows from the denial of one
alternative.

1 Page 71 seq.

DEDUCTIVE FORMS

59

The formal rules of such arguments as the above
are easy to understand. We shall, accordingly, state
these rules briefly, and shall adopt the convenient
device of using P to stand for any one proposition,
Q for any other proposition.

Hypothetical argument. This has two forms :
(I) If P, then Q. (II) If P, then Q.

not Q
not P.

Rule (1) : Affirm the Antecedent; the affirmation
of the Consequent follows.

Rule (2) : Deny the Consequent ; the denial of the
Antecedent follows.

It should be observed that nothing follows from the
denial of the Antecedent, nor from the affirmation of
the Consequent. This follows from the fact that the
Antecedent and the Consequent cannot be simply
interchanged. If it keeps fine, he will go out, is not
equivalent to If he goes out, it keeps fine. He may be
forced to go out, even though he dislikes wet weather.
Had the original proposition been Unless it keeps fine,
he will not go out, then the denial of the condition it
keeps fine would entail that he does not go out. But
in this case the antecedent is If it does not keep fine,
for ‘ Unless ’ means ‘ If . . . not ’. Hence, the denial
of fine weather constitutes an affirmation of this
Antecedent. The fallacy of Affirming the Consequent
(i.e. asserting the Consequent to be true and, thence,
concluding that the Antecedent is true) is very
common.1 The fallacy consists in the assumption that
the Consequent has only one condition.

1 For a further discussion of this fallacy, see p. 88 below.

60

LOGIC IN PRACTICE

(HI) Alternative Argument.

Either P or Q.
not P.

Q.

Rule : Deny one alternative ; the affirmation of
the other alternative follows.

Since the alternatives are not exclusive, nothing
follows from the affirmation of one alternative.

(IV) Disjunctive Argument.

Not both P and Q.

P.

not Q.

Rule : Affirm one of the disjoined propositions ; the
denial of the other follows.

The denial of one proposition in the disjunction
does not entail the affirmation or denial of the other.
Both may fail to be the case, since the premiss merely
asserts that at least one is not the case.

There are various complicated arguments consisting
of combinations of different compound propositions.
These, however, do not exhibit any other logical
principles than those with which we have dealt. We
shall notice one form only, the Dilemma, which is
familiar to educated people. Its use is primarily
rhetorical, since it affords an effective argumentative
device. The Dilemma is a form of argument consisting
of two hypothetical propositions conjunctively affirmed,
and an alternative premiss in which the antecedents
of the hypothetical premiss are affirmed or the
consequents are denied. One example will suffice :

If you succeed in your Recovery Campaign,”

a well-known American is said to have remarked

DEDUCTIVE FORMS

61

to Mr. Roosevelt, “ you will be known as America’s
greatest President. If you fail you will be known
as the worst.” “No,” Mr. Roosevelt is reported to
have answered ; “ if I fail, I shall be known as
the last President.” ’

This, again, is not an argument ; it presents the
material for an argument. We may formulate the
argument, abbreviating the statements :

If Roosevelt succeeds, he will be known as America’s
greatest President, and if he fails, he will be
known as America’s last President ;

But, either he will succeed, or he will fail ;
Therefore, either he will be known as America’s
greatest President, or as its last.

The form may be symbolized, using simple capital
letters to stand for simple propositions, as follows :

If P, then Q, and if not P, then R ;

But either P or not P ;
either Q or R.

In this form the Antecedents of the two hypothetical
propositions are contradictories ; hence, one of them
must be the case. This is not always so. We must
recognize four more forms :

If P, then Q, and if R, then T ;

But either P or R
either Q or T.

Another form is yielded by the alternative denial of
the Consequents, entailing the alternative denial of
the Antecedents. This form need not be separately
symbolized.

62

LOGIC IN PRACTICE

If P, then Q, and if P, then T ;

But P

Both Q and T.

Here the second premiss is simple, since the Ante-
cedents are common. The denial of Q or the denial
of T, entails the denial of P :

If P, then Q, and if R, then Q ;

But either P or R

/. Q.

Here the Consequents are common, and are asserted
to follow from both of two Antecedents.

The Dilemma is invalid if the alternatives are not
exhaustive. For example :

If we are marked to die we are enow
To do our country loss ; and if to live,

The fewer men, the greater share of honour

does not supply a premiss for a valid Dilemma.
Henry V neglected to notice that his army might
have been neither ‘ marked to die ’ nor ‘ marked to
live ’ ; the fortunes of the battle might depend upon
the numbers of the opposed armies.

The validity of a deductive reasoning depends upon
formal principles so easily apprehended that the plain
man may think them too obvious to need statement ;
such as the Principles considered in this chapter, viz.
Non-Contradiction, Excluded Middle, Applicative
Principle, Dictum de omni et nullo. No one who
understands these Principles is likely to deny them
to be true. But they are not the less important
because they are obvious. Unless the conclusiveness
of our reasonings could be shown to depend upon such

DEDUCTIVE FORMS

63

obviously true principles, we should have no good
reason for holding that our conclusions were conclusive.
As logicians we want to know what are the principles
which guarantee the validity of our reasoning. These
principles must be formal in virtue of the fact that
conclusive reasons in one case must be conclusive
reasons in any other case of the same form. If,
therefore, the conclusion of a valid reasoning is false,
its falsity must be sought in the material conditions ;
at least one of the premisses must be false. Every
deductive argument consists in drawing a conclusion
from a premiss, or a set of premisses which together
entail the conclusion. If the premisses are true, the
conclusion must also be true.

CHAPTER IV

AMBIGUITY, INDEFINITENESS, AND
RELEVANCE

‘ The line of all progress in disputes is towards definiteness —
definiteness of issue, definiteness in the conception of the facts
appealed to, and of the precise meanings of those facts.’

A. Sidgwick

IN discussing examples of reasoning in the last
chapter we did not pause to inquire whether we
clearly understood the sentences used to express our
statements. When we converse or argue with other
people, when we read or write — in short, on nearly
every occasion of reflective thinking — we use lan-
guage. A language consists of a set of symbols
capable of being combined in various ways in order
to express different states of affairs. Symbols are
signs used by some one, in accordance with a con-
vention, to refer to something. Words are one land
of symbols. They are sounds (in spoken language) or
marks (in written language) used by those who speak
the language. Words have meaning, but the sound,
or mark, has not meaning of itself ; it becomes a
sign, and thus acquires significance or meaning
through its use in accordance with a convention. It
is important to stress this conventional element since
it is apt to be forgotten. We then come to think of
the mark, or the sound, as the word, and suppose that
we can precisely determine the meaning of the word
by looking up the mark in the dictionary. Thereupon,
Ave fall into the mistake of supposing that words can

64

AMBIGUITY, INDEFINITENESS, RELEVANCE 05

be exhaustively divided into those which are ambi-
guous and those which are free from ambiguity. Both
these suppositions are mistaken. Meaning belongs
only to the sign as used. Hence, to know what a given
word means, we must know how it is being used in
the context in which the speaker1 is using it. Since
the contexts in which some words are used present
considerable similarity, these words have a compara-
tively fixed reference which enables us to speak of the
meaning of the word, e.g. ‘ table ‘ son ‘ member
of Parliament ’. This meaning may be ascertained by
consulting a dictionary.

We may become clearer about the nature of
meaning if we ask how a sign comes to be used as a
word having meaning. We may first ask what is
involved in understanding signs.2 Something has
already been said about this process in Chapter I.
We are now concerned with signs deliberately used
to signify something other than themselves, i.e. with
symbols. Motorists are familiar with various signs
giving them information about the road along which
they are travelling. They may see a sign f, which
indicates a sharp comer ahead by describing, to some
extent, the shape of the road. The verbal sign
‘ botar ar cie ’ would not indicate anything to a
motorist ignorant of the Irish language. But H may
indicate to him ‘ Junction to left if he is at all
familiar with the use of these diagrammatic signs.
Words do not for the most part imitate what they

1 Wherever we say ‘ a speaker ’ we could add ‘ or writer ’ ;
likewise, 4 hearer ’, or 4 reader ’. For brevity, we shall henceforth
speak of 4 the speaker ’. This is in conformity with the ordinary
usage of 4 speak ’ in the last sentence.

2 For a further discussion, see L. S. Stebbing : A Modern
Introduction to Logic, p. 12 seq.

5

66

LOGIC IN PRACTICE

indicate ; we have to learn that the mark son (and
the corresponding sound) stands for a male offspring,
generally of a human parent. We have to learn what
a flag at half mast signifies by observing the occasions
on which the flag is thus lowered. In the same way
we learn a language, not by paying attention to the
sound (or mark) as a sound (or as a mark), but by
attending to what it is some one is using the sound
(or mark) to refer to. A sign is understood when it is
known what it is that some one is using the sign to
signify. A hearer understands a word used by a
speaker when he is referred to that which the speaker
intended to indicate to him. Words, as Aristotle
pointed out, are ‘ sounds significant by convention
What the word is used to refer to may be conveniently
called its referend. When the indication fails, mis-
understanding results. Thus, if A (the speaker) says,

‘ Look at that queer thing ’, and B (the hearer) takes
‘ that queer thing ’ to refer to his cherished statue of
Buddha, whereas A was referring to a queer moth
fluttering round the lamp, communication has failed.
We say that B ‘ has misunderstood A ’ ; he has, in
fact, misunderstood what A says.

Owing to the fact that some signs are frequently
used with the same reference, dictionaries may help
us to discover howT a certain word is most commonly
used by those wiio speak the language. This is so in
so far as the word, wiiose meaning we seek to deter-
mine, is translated by a synonym which we already
understand, or a description is given in terms of wrords
already understood. We do not clearly understand
a wrord unless we could ourselves use the word in a
sentence the reference of which we understand. For
this reason, good dictionaries usually give examples

AMBIGUITY, INDEFINITENESS, RELEVANCE 07

of sentences in which the word is used. Understanding
words depends upon knowing the context.

We may take an example of learning a word which
was unfamiliar. Between 1924 and 1928 a new word
was used by writers on economics, ‘ rationalization \
We look it up in the dictionary and find the following :

‘ The scientific organization of industry to ensure the
minimum waste of labour, the standardization of
production, and the consequent maintenance of prices
at a constant level.’1 Provided that we understand
the words used in this description, we now understand
‘ rationalization ’ in the context of economic state-
ments. But we should hesitate to apply the word
until we have had some examples of rationalization ;
we shall understand better when we realize that the
word ‘ was coined to find a name for what was felt
to be a new phase in the history of the world economic
system ’,2 and have discovered what exactly that
‘ new phase ’ was. Other examples of words and
phrases used in new senses to refer to hitherto un-
described facts are ‘ complex ’ and ‘ unconscious
mind ’ as used in modern psychology ; ‘ inflation ’
and ‘ deflation ’ as used in economics.3

The importance of taking note of the context in
which a word occurs is very great. Words, as we have
seen, have a reference. Most words are descriptive,
i.e. they are used to refer to characteristics or proper-
ties, which may belong to something. For example,

1 Shorter Oxford English Dictionary.

2 G. D. H. Cole : The Intelligent Man's Guide through World
Chaos, p. 19. It is not strictly correct to say that the word was
‘ coined ’ ; it was adapted, no doubt for definite reasons of
association, on the ground that to give a dog a good name is to
beatify him.

3 It is curious that the economic sense of these two words does
not occur in 1933 edition of the Shorter O.E.D.

68

LOGIC IN PRACTICE

‘ bright red ’ is used to refer to any one of a range of
colours. To understand what ‘ bright red ’ means, we
must have actually seen something that is bright red.
‘ Bright red ’ is indefinite in its reference ; hence
people may not agree whether or not a given colour
is to be called ‘ bright red ’ or not. There are no
words which uniquely refer to shades of colours. All
descriptive words are more or less indefinite. A word
(or phrase) is indefinite when its reference is not
uniquely determined. Clearly indefiniteness admits of
degrees. We can sometimes achieve uniqueness of
reference by using a combination of words in such a
way that this combination could be used only to refer
to one thing, e.g. ‘ the colour of the covers of this
book as now seen by the speaker ’ ; ‘ the present
(1933) Prime Minister of this country £ the author
of Too True to be Good ’. The significance of the word
‘ the ’ is just to indicate uniqueness of reference. In a
context, uniqueness of reference may be secured by
the help of demonstrative gestures, actual pointing,
bodily presentment of something, and so on. In
speaking we are not talking about words, we are using
them to talk about something else, except in the
comparatively rare cases in which we are concerned
only with questions of language. That is why our
understanding of what is said depends upon the
whole situation in which the speaker is using the
words, i.e. upon the context. It is for this reason
that the indefiniteness of descriptive words does not
prevent us from using them in such a way that they
have uniqueness of reference.

The referend of a sign used as a descriptive word is
a characteristic, or a set of characteristics. When any
one of a number of somewhat different characteristics

AMBIGUITY, INDEFINITENESS, RELEVANCE 69

could be properly referred to by the same word
then this word is, more or less, indefinite. Some
words are not only indefinite, they are also vague.
A word, or phrase, is vague when it is so used that
we could not tell in a given situation whether or not
the word was applicable. Such words as ‘ bald ’,
‘ fat ‘ successful business man ’, ‘ security ‘ value ’,
are vague. We cannot make a precise distinction
between the state of a man’s head which justifies us
in calling him ‘ bald ’ and the state which would more
properly be described as ‘ having very little hair
Likewise with the other examples. A certain degree
of vagueness in descriptive words is often quite
unimportant for the ordinary purposes of life. More-
over, some words are properly vague since they are
used to refer to a characteristic admitting of con-
tinuous variation. This is the case with ‘ bald ’ and
with ‘ intelligent ’, and with all such words as ‘ idiot ’,
£ imbecile ‘ insane ’. Intelligence can be manifested
in various degrees ; it is impossible to draw a sharp
line between those who possess intelligence and those
who do not. It is an error in good sense to insist that
a speaker should draw a sharp fine where, in fact, no
such fine can be drawn. To admit this is not to deny
the important difference between being intelligent and
being unintelligent ; it is to admit that what exactly
are the characteristics indicated by ‘ intelligent ’
cannot be precisely determined. An error of an
opposite kind would be committed if we argued that
since no sharp fine can be drawn between the intelli-
gent and the unintelligent, there is no difference
between them. We shall see the importance of these
considerations when we come to deal with the nature
and utility of definition.

70

LOGIC IN PRACTICE

Indefiriiteness and vagueness must be carefully
distinguished from ambiguity. We have seen that it
is not necessarily a defect in descriptive language that
it should be more or less indefinite and more or less
vague. The ambiguous use of words is always vicious.
A word is used ambiguously when the same word is
used to indicate different referends without the
speakers realizing that there is a difference in what
is referred to. It is only in a context that ambiguity
can arise. A word considered in isolation could not
properly be said to be ambiguous. Some examples
may make this point clear. The sign rationalization
might be used with four different meanings : (1) in
economics, to indicate what was given above on
page 67 ; (2) in mathematics, to indicate the process
of clearing from irrational quantities ; (3) in modern
psychology, to indicate the assigning of incorrect
motives in explanation of a person’s behaviour ;
(4) in the original use of the sign, to indicate making
rational or intelligible. No doubt there are sound
historical reasons why the same sign should be used
with four such different indications, and we need not
in this book discuss whether or not wTe have not only
the same sign but also the same word. The point is
that ‘ rationalization ’ is not ambiguous. No one
could be so stupid as to use it in two (or more) of
these different ways without knowing that he had done
so. To use a word ambiguously is to be confused with
regard to its different indications. It is very easy to
be thus confused. Ambiguity is prevalent because
our thinking is so unclear. One of the most important
tasks of the practical logician is to try to point out
some of the various ways in which vre fail to think
clearly because wre have failed to notice a shift in the

AMBIGUITY, INDEFINITENESS, RELEVANCE 71

reference of the words we use. It would take a whole
volume, much larger than this little book, to deal
at all properly with this topic. All that is possible
here is to select a single set of words often used
ambiguously, in the hope that the reader may then
be led to notice other examples for himself. We need
not waste time pointing out the difference of reference
indicated by ‘ a bald head ’ from that indicated
by £ a bald statement ’, nor the difference between
‘ a fair bargain ’ and £ a fair complexion nor be-
tween £ a General Strike ’ and £ a strike on the
head

It is in discussions concerning politics, economics,
religion, education, and art that ambiguity is most
prevalent and most harmful. This is only to be
expected. Where the subject is complicated, our
thinking is likely to be confused ; where our ordinary
and passionate interests are concerned, we are likely
to accept without much scrutiny any argument
defending a position we want to hold. In such cases
we may fail to notice a shift in meaning of the words
used. Discussions concerning the General Strike of
1926 afford an amazing crop of ambiguities, as the
following quotations will show : 1

£ Constitutional Government is being attacked.

. . . Stand behind the Government . . . confident
that you will co-operate in the measures they have
undertaken to preserve the liberties and privileges

1 These quotations (abridged) are taken from Leonard Woolf’s
After the Deluge, Vol. I, p. 304 seq. The italics are mine, and are
designed to call the reader’s attention to certain words which
appear to me to be used ambiguously by the speakers. The
reader should bear in mind that we are here concerned with an
example of confused thinking resulting in ambiguous language ;
we are not concerned to take sides on the issue discussed.

72

LOGIC IN PRACTICE

of the people of these islands. The laws of England
are the people's birthright ’ (Mr. Baldwin).

‘ A General Strike, such as that which it is being
sought to enforce, is directly aimed at the daily life
of the whole community ’ (Lord Oxford and Asquith).

‘ This General Strike was not a strike at all.
A strike was perfectly lawful. . . . The decision of
the Council of the Trade Union Executive to call
out everybody, regardless of the contracts of those
workmen they called upon, was not a lawful act at
all ’ (Sir John Simon).

‘ The plain fact was that, not as a matter of
narrow law, but as a matter of fundamental consti-
tutional principle, when once they had a proclama-
tion of a general strike such as this, it was not,
properly understood, a strike at all. A strike was a
strike against employers to compel employers to
do something. A general strike was a strike against
the general public to make the public, Parliament, and
the Government do something 5 (Sir John Simon).

A careful examination of these statements will show
how confused thinking is revealed in the shifting
meanings of the italicized phrases. Are not the
strikers, we may well ask, to be included among ‘ the
people of these islands ’ ? Do they not belong to the
‘ community ’ ? Is a general strike not a ‘ strike ’ at
all ? What does Sir John Simon mean by ‘ properly
understood ’ ? An examination of the speeches and
writings made by various supporters of the Govern-
ment, in May and June 1926, will show how uncertain
was the reference intended by the words ‘ legal ’,
‘ war ’, ‘ enemy ’, as used on both sides. A Conserva-
tive, for instance, tends to identify ‘ the community ’

AMBIGUITY, INDEFINITENESS, RELEVANCE 73

with ‘ the middle classes ’, and ‘ the Government ’
with ‘ the State ’ ; a trade unionist may identify ‘ the
community ’ with ‘ the workers ’, and so on. We may
well ask — as Prof. Laird asked, in a different con-
nexion— ‘ When the good of “ the ” community is set
before us and proclaimed to be the consummation of
all our loyalties, it is reasonable to ask, What com-
munity ? ’ 1 To see the necessity of asking this
question is to realize how easily we may be misled
through a failure to recognize that a word is being
used ambiguously.

A final example may be given of confused thinking
resulting in a failure to recognize that language is
being used both ambiguously and with an improper
degree of vagueness. A writer in The Spectator
(December 30th, 1932) said :

‘I do not believe in the possibility of eliminating
the desire to fight from humankind because an
organism without fight is dead or moribund. Life
consists of tensions : there must be a balance of
opposite polarities to make a personality, a nation,
a world, or a cosmic system such as God planned.’

The writer gives as his reason for the conclusion —
that it is not possible to eliminate the human being’s
desire to fight — that 1 an organism without fight is
dead or moribund ’. The word ‘ fight ’ is familiar,
and the wnriter has failed to notice that the reference
has shifted in the conclusion. In his premiss ‘ fight ’
is used in the sense of ‘ struggle against the environ-
ment ’, i.e. tension, in the sense of a conative urge, or
drive, towards something not realized. It may be
true that without a balance of ‘ tensions ’, there would
1 A Study in Moral Theory, p. 244.

74

LOGIC IN PRACTICE

be no developed personality. But in the sense in
which ‘ fighting ’ means ‘ being at war ’ — which is the
sense required for the conclusion — it does not follow
from the fact asserted in the premiss that human
beings must continue to desire to fight in order to
maintain their personality. It may be doubted
whether the writer had any clear conception as to
what exactly was the evidence upon which he was
attempting to base his conclusion. Certainly his
conclusion may be true, but his argument fails to
support it.

It is important if we wish to think clearly to be
constantly on guard to see that there is no shift of
reference in the course of an argument. Logicians
have been wont to insist that the middle term of a
syllogism must not be ambiguous. If the middle term
has one reference in one premiss, and a different
reference in the other, then, although there may be
only one word, there are two terms, and hence no
middle term. The middle term just is that term which
is the same in both premisses. It is the essential
function of the middle term to secure that the pre-
misses have a point of identical reference. This is
the reason why the middle term must be distributed.
If we were thinking only about symbols we could secure
identity of reference, and thus freedom from ambi-
guity, by putting the right symbol in the right place.
We should thus avoid undistributed middle, whilst
‘ Y ’ would appear in both premisses. But what looks
the same word may not have the same reference. In
the symbolic form this danger of ambiguity is con-
cealed. Nor is it confined to the middle term of a
syllogism ; a term in the conclusion may fail to
indicate what was indicated by the corresponding

AMBIGUITY, INDEFINITENESS, RELEVANCE 75

term in the premiss. This danger is both prevalent
and insidious. It is so easy to attribute to words the
fixity of symbols. Consider the argument : ‘ Of
course Christians must seek peace, and not war.
Christians are followers of Christ, and those who
follow Christ certainly seek peace.’ It is not at all
unlikely that the middle term of this syllogism is not
used with the same reference in both premisses ;
possibly, also, ‘ Christians ’ does not indicate in the
conclusion what it was used to indicate in the premiss.
It may even be the case that ‘ Christians are followers
of Christ ’ may be a verbal proposition, viz. a proposi-
tion stating what a word means. If so, the conclusion
merely re-states the other premiss. If not, the
possibility of serious ambiguity remains. It might be
replied that the speaker means ‘ true Christians ’.
The addition of this qualification is by no means un-
usual. Its tendency is to beg the question. This fallacy
is so prevalent that a little must be said about it here.

To beg the question is to assume the point at issue ;
in a (faultily) reasoned argument, the fallacy may
take the form of using as a 'premiss the conclusion
which the argument purports to prove. Perhaps we
do not often commit the fallacy in the gross form in
which it was committed by one of Jane Austen’s
characters. Unfortunately, the passage is too long to
quote in full. It must suffice to quote the following :
‘ “ Let me explain myself clearly ; my idea of the
case1 is this. When a woman has too great a propor-
tion of red in her cheeks, she must have too much
colour.” “ But, Madam, I deny that it is possible for
anyone to have too great a proportion of red in their

1 The case being whether, or not, Mrs. Watkins had too much
colour.

76

LOGIC IN PRACTICE

cheeks.” “ What, my Love, not if they have too
much colour ? ” 51 Here there is no ambiguity. Both
speakers probably understood the same by ‘ red ’ and
by ‘ colour Yet, if so, how could the question have
been so flagrantly begged ? Perhaps the reader will
think this discussion too stupid for ordinary life. It
would not, however, be difficult to find equally glaring
instances, probably in one’s own reasonings, certainly
in those of others. It is true that our begging of the
question is usually less obvious owing to its being
cloaked by unclear, ambiguous, vague language. That
is why this fallacy may be fittingly dealt with in this
connexion.

Let us return to the emendation of a challenged
conclusion, by means of the qualification ‘ A true so
and so For example, we make a sweeping generali-
zation about, say, musicians and Wagner’s operas, to
the effect that ‘ No musicians nowadays admire
Wagner ’. When challenged, the speaker may reply,

‘ Well, no true musician does Pressed to make
definite the distinction between ‘ a true musician ’
and just a ‘ musician ’, he might fall back, ultimately,
on the test of admiring Wagner as differentiating the
pseudo-musicians from the ‘ true ’ ones. In so doing,
he would beg the question ; his fallacy might be
concealed from himself because he had no clear
conception of the reference of the term ‘ musician ’.
He has used it with an improper degree of vagueness,
otherwise he would have seen (we may hope) that,
since — in his view — the appropriateness of calling any
one a musician depends upon his attitude to Wagner,
it would be merely verbal to say that musicians have
this attitude.

1 MS. Volume the First.

AMBIGUITY, INDEFINITENESS, RELEVANCE 77

No precise rules can be laid down to enable us to
determine whether a given word is being used ambi-
guously, or with an improper degree of vagueness.
There are no principles which could guide us in
avoiding ambiguity. Only in a context is a word
ambiguous. That is why symbols — such as the X,
Y, Z we have used — are unambiguous ; they are cut
free from a context. In this abstraction from a
context lies the value of symbols in revealing the
formal conditions ; but therein lies also their limita-
tion from the point of view of the material conditions
of reasoning. The only advice that can be offered is
to be on the look-out for ambiguities. The habit of
asking certain questions is a help. If we ask what
must also be the case if what we are saying is true,
then we may notice that what we say admits of
different interpretations. Again, we may ask our-
selves whether we are dealing with exceptional cases,
e.g. with cases which do not quite fall under the
adopted usage. It may be generally correct to say
that a certain characteristic is associated with other
characteristics referred to by a given word, e.g.
‘ religion but it may nevertheless be incorrect in
this case so to associate it. Thus, it might be main-
tained that religion is good because it involves
worship. But it may be relevant to ask whether
worship of any god, or thing, is good, or only of a
god having such and such characteristics. One
person’s ‘ god ’ is another person’s ‘ devil This
point may be expressed symbolically. It may be true
to say genera'ly that X is Y, whilst in some special
case that which is quite correctly called ‘ X ’, is yet
not Y. An obscure perception of this divergence is
often responsible for our taking refuge in such

78

LOGIC IN PRACTICE

qualifications as ‘ a true patriot ‘ a true Liberal
* a schoolboy as such \

The reader might suppose that the insidious danger
of unclear language could be overcome if we were to
define the words we use. It would, however, be a
mistake to expect much help from such a practice,
useful though it may be at times. Space is lacking
to deal adequately with the nature of definition. It
must suffice to point out that to be able to define a
word is already to know what it signifies. In defining
a word we set forth certain characteristics which may
belong to whatever the word is correctly used to refer
to. Words are defined by means of other -words, and
ultimately by words which do not stand in need of
further definition. Defining is not primarily a process
of making our own thought clear ; it is the signal
that clarity has been achieved. A wiser person may
help us to think more clearly by showing us how a
word should be defined ; but we cannot lift ourselves
out of a muddle by jumping to a definition. For this
reason the rules of definition, which it is customary
for logicians to lay down, are not of practical use.
These rules provide that a definition must not be too
wide, nor too narrow, nor expressed in obscure or in
figurative language. But the difficulty just is to know
what would be too wide, or too narrow. For example,
are we to define ‘ liberal ’ (as used with reference to
a political party) in such a way that it includes the
followers of Sir Herbert Samuel and Sir John Simon
and Lloyd George ? Or are any of these to be
excluded ? A logician’s knowledge of formal rules
affords no help.

Nevertheless, there is a stage in most argumentative
discussion at which precise definition is required,

AMBIGUITY, INDEFINITENESS, RELEVANCE 79

whilst the search for a satisfactory definition may be
itself enlightening. A good example of the need for
clearly defined terms was provided in recent dis-
cussions concerning spending and saving. Broadcast
talks by Mr. J. M. Keynes and Sir Josiah Stamp
revealed much misunderstanding as to what exactly
constitutes ‘ spending ’ and 4 saving ’ respectively.1
Numerous letters to The Times showed how wide-
spread the confusion was, and how opposite con-
clusions appeared to be drawn from the same premisses
owing to the fact that the terms employed were
unclear in their reference. Mr. Keynes argued that
every pound saved put a man out of work, so that
saving was not economically justifiable in a time of
unemployment. Sir Josiah Stamp urged that habits
of thrift were essential in a time of economic depres-
sion. He was, however, led to the conclusion that
4 true saving is only another way of spending, and
employs just the same Sir Josiah Stamp, then,
distinguished between 4 saving ’ in the sense of
4 hoarding and 4 saving ’ in the sense of 4 investing ’,
i.e. 4 spending upon a different set of objects ’. This
prolonged discussion would have been considerably
clarified had the disputants explicitly defined the
terms used, and distinguished between different sorts
of saving and expenditure. The common reader is
not helped in doing his duty as a citizen when he is
advised merely that 4 saving ’ creates unemployment
but 4 true saving ’ benefits the community. The
correspondence in The Times showed that it is easier
to advise people to 4 save wisely ’ than to show
wherein lies the distinction between wise and univise

1 See The Listener, January 11, 1933; January 14, 1931;
January 28, 1931.

80

LOGIC IN PRACTICE

saving.1 Clearly to understand this distinction is to
know what differentiates one sort of saving from
another sort.

Distinguishing between different sorts of the same
fundamental kind constitutes logical division. Space
permits only a few words on this topic. 2 We under-
stand the characteristics referred to by a general term
(e.g. Saving, Liberal, Ship) when we are able to specify
its ramifications. Thus, for example, we may divide
Spending into (1) Hoarding, and (2) Investing. We
may subdivide Investing into (i) Immediate Invest-
ments, and (ii) Delayed Investments ; (i) may be again
subdivided into (a) Investments aiming at the direct
benefit of the investor or his descendants, (b) Investments
aiming at socially useful results. This division is not
complete, but it may suffice to show that a fuller
understanding of the nature of a given class may be
attained when we have distinguished its various sub-
classes (or, as they are called, species), and the various
sub-classes of these sub-classes. The class which is
subdivided is called a genus, relatively to the species
into which it is divided. The fundamental charac-
teristics of the original class, or genus, must be present
in each sub-class ; it is these generic characteristics
which justify the use of the same class-term. The
specific characteristics distinguishing one sub-class
from another, may justify a distinction between one
species and another, e.g. between wise and unwise

1 The conclusion of the Broadcast discussion between Keynes
and Stamp affords a striking instance of the futility of an economic
discussion in which the fundamental terms are not defined.
Sir J. Stamp closed the discussion with the remark : ‘ In short,
this saving and spending of ours are really, or ought to be, sort
of sister shows.’

2 For further discussion of Logical Division, see L. S. Stebbing :
A Modern Introduction to Logic, Chap. XXII.

AMBIGUITY, INDEFINITENESS, RELEVANCE 81

spending. Logical division must proceed on an
orderly basis, i.e. there must be a single principle
upon the basis of which one species is differentiated
from a co-ordinate species. The co-ordinate species
must be exhaustive of the wider class, or genus, within
which they fall ; otherwise, certain members of the
genus will not have been included. A well-arranged
library exhibits a logical division, for in it, no books
(belonging to the library) will have been omitted,
whilst no book will be in two places at once. As the
catalogue of a library suggests, the principle of division
is relative to purpose. We may divide books accord-
ing to authorship , date of publication , subject-matter,
binding, etc. Each principle of division would yield
a different arrangement of the classes concerned.
What would, in given circumstances, be the most
fruitful division depends upon what differentiating
characteristics are most relevant to the question at
issue. A logical division is fruitful when it gives rise
to inferences relevant to the topic under discussion,
i.e. when, from knowing the place of a sub-class in
the orderly arrangement of classes, we can infer how
its members resemble and differ from the members of
other sub -classes.

The reader will have noticed that making a satis-
factory logical division depends upon our knowing the
material conditions, so that, here again, formal rules
do not afford much practical help, except in so far as
they may aid us to recognize how a proposed division
fails to be sound. It is easy to insist upon the need
for relevance ; it is often difficult to know what is
relevant. A wise man will not attempt to argue about
a subject on which he is ill-informed. Unfortunately,
many of the topics on which we hold strong opinions
6

82

LOGIC IN PRACTICE

are topics concerning which we are sadly ignorant.
This ignorance of relevant considerations leads us to
construct faulty arguments and renders us a prey to
the unscrupulous disputant. A common fallacy of
this type is that of the irrelevant conclusion,1 which
consists in establishing a conclusion which is other
than the conclusion intended to be proved. For
example, a disputant may attempt to throw doubt
upon an opponent’s statement by asserting that it is
to the advantage of the opponent to believe in the
truth of his statement. This, however, is not the
point at issue ; hence, the argument is irrelevant
unless it can be shown that the opponent’s sole reason
for accepting the statement is his desire for it to be
true. It is by no means uncommon for conservatives
to argue that socialism must be unsatisfactory since
it is based upon the envy of the ‘ have-nots ’ for those
who have ; socialists, on the other hand, sometimes
seem to suppose that it is a sufficient argument
against capitalism, to show that capitalists desire to
keep what they possess. Another form of irrelevant
argument consists in blackening the characters of
those who support a proposition ; still another form
is found in ridiculing the supporters. A joke often
provides an excellent diversion, but it is a diversion,
a turning away from the point at issue. The forms
which irrelevant arguments may take are too numer-
ous to be dealt with here. One more, very common,
form may be noted. In a recent trial, the counsel in
defence of a convicted prisoner, sought to mitigate
his sentence by calling the judge’s attention to the
fact that the prisoner had a wife and five children.

1 This is known as the fallacy of ignoratio elenchi (i.e. ignoring
the point at issue).

AMBIGUITY, INDEFINITENESS, RELEVANCE 83

This is clearly an irrelevant argument. If, however,
the counsel were to plead that the prisoner’s previous
record had been good, and that he had, apart from
this lapse, dealt honourably in business, then his
argument would have been relevant. The form of an
irrelevant argument is : You must accept Q because
you accept P, where, in fact, P does not establish Q.
Thus formally stated it might be supposed that no
honest thinker could be so misled. The difficulty is,
however, that our arguments are not set out briefly,
in clear language, and consequently we easily fail to
perceive the want of connexion between P and Q.
The use of emotionally charged language may create
an attitude of mind which makes us accept an unsound
argument. Thus we find socialists accusing capitalists
of ‘ robbing ’ the poor ; we find capitalists dubbing
unemployment insurance as ‘ the dole ’ ; we think of
our enemies (in war) as ‘ murderous foes and of our
own men as ‘ heroes ’. The reader will be able to
supply other examples. The only way to avoid being
led into unclear thinking of this kind is to attempt
to translate language directly arousing emotional
attitudes into plain speech, and to consider whether
the reasons urged against one’s opponent would be
relevant against oneself. We are all inclined to the
fallacy of ‘ special pleading i.e. accepting (or
refusing) in one’s own case an argument which one
refuses (or accepts) on the other side. For example,
a person may condemn the ‘ dole ’ on the ground that
the recipient has not earned it by work, whilst
accepting the view that those who inherit wealth may
five on an unearned income.

A closely allied fallacy consists in asserting an
indisputable contention and thence proceeding to

84

LOGIC IN PRACTICE

another proposition in no way related to the former.
The hearer accepts the platitude, and may fail to
notice that the contention in dispute is in no way
established. For example, it may be argued that
‘ human beings are governed by primary human
instincts, not by socialist theories ’, and that therefore
socialist theories are wrong. This therefore is a non
sequitur. Socialist theories may be profoundly mis-
taken, but it is not in such a way that they can be
disproved.

It is useful to cultivate the habit of asking oneself
whether a given statement is supported by the argu-
ment offered. If so, the premisses must be consistent
with the conclusion, and must provide some reason
for it. This reason will, we have seen, be valid in any
other argument of the same form. The language used
must be tree from ambiguity ; the point at issue must
be definite. We do not disprove a proposition by
showing that the argument offered in its support is
unsound, but, unless we are offered another, and a
sound, argument in support of it, we have no reason
for accepting it as true.

CHAPTER V

THE ESTIMATION OF EVIDENCE

' A straw will show which way the wind blows.’ — Old Saw

THE connexion between a mass of evidence and
that which it evinces may be approached from
two different points of view. The first is exemplified
in the attitude of legal counsel, for the defence, or for
the prosecution ; the second, in the attitude of a
detective attempting to discover the man who did the
deed. For brevity, we may refer to the mass of
evidence as the data, and to that which it evinces as
the probandum. The counsel accepts the probandum
as already determined ; his problem is to select from
miscellaneous, and possibly conflicting data, just
those facts which point to the already accepted
probandum. The detective seeks a probandum which
is, at the outset, completely undetermined ; his
problem is to determine the probandum by examining
the data, selecting what is relevant, and recognizing
its significance. His selection is guided by an hypo-
thesis, more or less capable of explicit formulation.
His thinking involves the three steps mentioned in
Chapter I ; if the conditions constituting the problem
are at all complicated, he may need to try out several
hypotheses before he is satisfied that he has hit upon
the correct solution. The detective’s task is more
difficult than that of the counsel. The data may point
in many different directions ; at first sight it may
even be the case that no definite probandum appears
to be indicated by the available data. Once the case

85

86

LOGIC IN PRACTICE

is completed, the detective, no less than the counsel,
may present his conclusions in deductive form.
Nevertheless, his reasoning remains essentially induc-
tive. For example, the detective may argue : ‘A’s
boots fit these footprints in the flower-bed ; there-
fore, A made these footprints.’ The cogency of this
argument depends upon the assumption that the soil
of a flower-bed will always respond in the same way
to pressure of a certain kind. The detective relies
upon his commonsense knowledge of the way things
happen : he assumes that there are uniformities of
behaviour and causal connexions. Or, he relies upon
somewhat more expert knowledge to the effect that
no two people have the same finger-prints. The
warrant for this assertion is to be found in the observed
fact that every person tested for finger-prints is found
to have peculiar markings. It should not be necessary
to multiply illustrations of the contention that the
significance of the observed facts — constituting the
original data — is wholly due to our knowledge of the
regular ways in which one happening is connected
with other happenings. The cogency of the counsel’s
argument depends likewise upon the previous accep-
tance of premisses derived from generalizations based
upon the assumption of uniform happenings.

It is customary to distinguish three modes of
inductive inference, viz. analogy , generalization, cir-
cumstantial evidence. Although the distinction is
useful up to a point, yet the three modes are of
fundamentally the same nature. Each of them is
based upon the recognition of relevant resemblances
and relevant differences. Inference by analogy con-
sists in inferring that, since two cases are alike in
certain respects, they will also be alike in some other

THE ESTIMATION OF EVIDENCE

87

respect. For example, since Mars resembles the Earth
in certain respects, we infer that Mars also is inhabited.
This may be a very risky inference, for Mars differs
from the Earth in some respects, and these differences
may be relevant to the property of being inhabited.
If so, then whatever may be the extent of the resem-
blance between Mars and the Earth, this resemblance
is unimportant from the point of view of the given
inference. Any respect in which Mars resembles the
Earth (e.g. revolving round the Sun) puts Mars into a
class consisting of at least two members, viz. Mars
and the Earth. This resemblance may then be the
basis of a generalization.1 Since the members of any
class resemble each other in some respect and differ
in others, the argument from resemblance must be
controlled. Hence, we are led to distinguish between
essential (or important) and unessential (or unimpor-
tant) resemblances and differences, and thus to form
classes. Generalizations relate to classes, and are thus
based upon analogy. We resort to simple analogy
(resemblance between individual instances) only when
the circumstances are too complex, or the case too
rare or too unfamiliar, for us to be able to fall back
upon the generalization invoked in the recognition of
a class.

The phrase ‘ circumstantial evidence ’ is most
usually employed to designate the form of reasoning
in which a set of evidentiary facts cumulatively point
to a certain definite conclusion although no single fact
itself suffices to indicate that conclusion. This is the
form of reasoning employed by detectives — at least
in detective novels. Poe’s Rue Morgue affords a good
example ; the committee of investigation — discussed

1 See p. 19 above.

88

LOGIC IN PRACTICE

in Chapter I — provides another example. The distin-
guishing characteristic of this mode of inference lies
in the cumulative force of a set of facts taken together.
It would be a mistake to suppose that this mode of
inference is confined to criminal investigation. On
the contrary, all reasoning of the form — If F1 and F 2
and F 3 and . . ., then P; but F1 and F 2 and F3
and . . . ; therefore P — falls under this mode.1

It should be observed that inference from circum-
stantial evidence involves generalization, and therefore
analogy. We saw this to be so in the case of arguing
about footprints. To infer that Fx indicates P is to
rely upon certain general characters of Fx as relevant
to P. In short, the significance of each separate fact
— i.e. each separate item in the evidence — depends
upon the thinker's knowledge of uniform behaviour,
i.e. of regular modes of happenings. The inference
as a whole is not a generalization, because no single
generalization would cover all the facts. We rely
upon circumstantial evidence when we are investi-
gating a unique occurrence ; since the occurrence is
unique no general rule could be formulated. Conse-
quently such an inference is not syllogistic, nor,
indeed, deductive at all. To regard it as deductive
would be to admit that the argument is fallacious,
since the consequent is affirmed. The conclusion,
however, is not that the set of facts entail P ; it is
that they indicate P. Hence, P may be false, not-
withstanding that the facts are as reported. The
inference accordingly, is inductive.2

Inference from circumstantial evidence is often
regarded as a chain argument. But if a chain is not
stronger than its weakest link, then this description
1 See pp. 25-26 below. 2 See p. 20 above.

THE ESTIMATION OF EVIDENCE

89

is inept. A single fact, F1? may weakly suggest F,
yet the strength of the cumulative evidence may be
considerable. Its strength is due to the consideration
that P alone fits all the facts. The weapon with which
the murder was done may belong to A, and A may
have had a motive to commit the murder, and may
have had the opportunity, and yet the murderer
might be B. It is certainty true that in real hfe, as
well as in detective stories, it may occasionally happen
that a person is entangled in a web of circumstantial
evidence pointing to the conclusion that he has com-
mitted a crime of which he is, none the less, innocent.
Even assuming that the difficulties of ascertaining
the relevant facts and of obtaining reliable evidence
from eye-witnesses have been overcome,1 we are forced
to admit that circumstantial evidence cannot suffice
to yield a certain conclusion. In inductive inference
we are never in a position to maintain that no other
conclusion is consistent with the evidence. But if all
the facts point to P, and no alternative possibilities
are discovered, then we feel it would be stretching the
‘ long arm of coincidence ’ too far, to reject P on the
ground that some unthought- of alternative would
explain away the set of facts which together indicate
P. A proper discussion of this topic would take more
space than is at our disposal. It must suffice to point
out that the inference is more reliable in proportion
as there is a reasonable probability that (1) each of
the accepted facts is adequately fitted into P ;
(2) no relevant facts have been overlooked ; (3) if
there had been contradictory facts they would have
been noted. These are big provisoes. Nevertheless,

1 On the difficulty of obtaining such reliable evidence, see
A. W. P. Wolters : The Evidence of our Senses, Chap. IV.

90

LOGIC IN PRACTICE

we may sometimes have reasonable confidence that
the facts warrant the conclusion.

How, we may ask, is this reasonable confidence to
be secured ? In the last resort only by rel}dng upon
our knowledge of uniformities of behaviour. In this
book we take for granted that we do have knowledge
of uniform connexions, as well as of happenings that
are mere coincidences, i.e. which are such that some-
times a happening of a certain kind is conjoined with
a happening of another kind, whilst sometimes these
two kinds of happenings are not conjoined. For
example : iron rusts, lead does not ; wood burns
easily, concrete does not ; arsenic is poisonous, lemon
juice is not ; Hitler is sometimes in a train, sometimes
in a house. In ordinary fife, all our reasoning pro-
ceeds upon the basis of knowledge of uniform con-
nexions. Our previous discussion of classes and of
generalization should have made this clear. Whenever
we ask such questions as what caused this fire, why
does iron rust, we are in the position of detectives.
Our witnesses are the observed occurrences ; we
cross-examine them by experimental testing. Instead
of asking whether A murdered B, we ask whether
arsenic is poisonous. To say that arsenic is poisonous
is to say that under certain circumstances arsenic
causes death.

In attempting to discover the causal properties of
things, i.e. the way a given thing behaves in relation
to other things, we rely upon certain principles of
discovery. These principles are derived from the
fundamental notion of causation. Two happenings,
X and Y, are causally connected when X is a necessary
and sufficient condition of the happening of Y.
Hence, the introduction of X into a situation will be

THE ESTIMATION OF EVIDENCE

91

accompanied by the happening of Y ; the removal of
X by the disappearance of Y. X is said to be the
cause ; Y is said to be the effect. Two principles
follow directly from the nature of a cause, i.e.
(1) Nothing is the cause of an effect which is absent
when the effect occurs ; (2) Nothing is the cause of
an effect which is present when the effect fails to
occur. Accordingly, in seeking for the cause of an
occurrence, Y, we shall look for situations in which Y
is present, and for situations resembling the former
in many respects but differing from them in the
absence of Y. These principles yield two derivative
principles, which may be called, respectively, the
Principle of Agreement and the Principle of Difference.
Two examples may suffice to show how these principles
are used.

A certain man finds that on eight successive
Tuesdays he has a headache ; but on no other days
during those weeks has he had a headache. He asks
what has happened on the Tuesdays which has not
happened on other days. He remembers that on
each of those days he has returned from the City by
the Underground Railway, whereas it is his usual
custom to return by bus. But on those Tuesdays he
had an early after-dinner engagement to play chess
at a friend’s house, and to get there in time he had to
travel by the quicker route. On other evenings he
plays chess in his own house, sometimes with this
friend, sometimes with others ; he does not then
need to be home earlier. On these days he does not
have a headache. The journey by Underground is
common to all the Tuesdays and nothing else seems to
be both common and peculiar to the days on which
he gets a headache. He therefore concludes that the

92

LOGIC IN PRACTICE

journey in the Underground is causally connected
with his headache. In reaching this conclusion he is
employing the Principle of Agreement. The con-
clusion is by no means certain. Yet, if he has played
chess with the same friend on other occasions when
he did not have a headache, and on these occasions
he had travelled home by bus, then the Principle of
Agreement makes it reasonable to suppose that the
Underground journey is responsible. If, further, he
came home by Underground one day and did not
go to his friend’s house, nor play chess, then the
probability that the conclusion is correct is
strengthened, since the cause must be present when
the effect is.

A healthy man eats a liqueur chocolate. Almost
immediately he falls down dead. It is concluded that
he was poisoned by what he had just swallowed.
This conclusion is reached by an application of the
Principle of Difference. At one moment the man is
alive and well ; a few moments later he is dead.
Nothing appears to have happened except the eating
of the chocolate ; hence, no other factors can be
responsible. If it is then found that cyanide of
potassium had been put into the chocolate, then we
shall be confident that this poison caused his death.
No doubt we should then reason deductively as
follows : Whoever swallows a certain amount of
cyanide of potassium dies immediately ; this man has
swallowed such an amount of cyanide of potassium ;
therefore he dies. The reader should observe, how-
ever, that no one would examine the chocolates to
see if they were poisoned unless it had been assumed
that the eating of the chocolate were an indispensable
condition of this man’s death. Many people eat

THE ESTIMATION OF EVIDENCE

93

chocolate and continue to live. Why, then, should
the chocolate have been examined ? The reason is
that the eating of the chocolate was the sole new
factor introduced into the situation in which the man
had been alive and healthy. Most people now know
that cyanide of potassium is poisonous, and that
chocolate is not. But at one time it was a discovery
that this property belonged to cyanide of potassium.
This discovery could be made only by noticing what
happened when cyanide of potassium was absorbed
by a living organism. If no other factor in the situa-
tion had been changed, then the Principles of Causa-
tion justify us in concluding that this additional
factor was the cause of the observed effect. The
condition that only one factor has varied is of great
importance. Neglect of it is partly responsible for
the very common fallacy of post hoc ergo propter hoc,
i.e. the fallacy of concluding that what has imme-
diately preceded an occurrence is the cause of that
occurrence. For example, a man curses his enemy,
who shortly afterwards dies ; there is a ‘ change in
the moon ’ and then a change in the weather. To
argue that the second (in either case) is causally
consequent upon the first is to mistake a temporal
conjunction for a causal connexion. We are tempted
to fall into this fallacy when one or other of the two
occurrences is especially striking. We cannot even
argue from a constant conjunction of two occurrences
to a causal connexion ; we require to observe a situa-
tion in which one factor can be eliminated. The
fallacy of post hoc ergo propter hoc is responsible for
many popular superstitions. The man who trusts to
his mascot to help him win a match may never have
tried what would happen if he left it at home.

94

LOGIC IN PRACTICE

The two examples previously given should suffice
to show that the discovery of causal connexions
depends upon an analysis of a complex situation.
Certain features of the situation must be simply
judged to be irrelevant.1 The man who had a head-
ache on Tuesdays would judge Tuesday as such to be
irrelevant ; the day of the week is important only in
relation to how its occupations differ from those of
other days. The colour of the carpet upon which the
poisoned man was standing would also be judged
irrelevant, since, presumably, he and other people
had stood on it before without ill-effect. It is, how-
ever, easy to rule out as irrelevant factors which are
indispensable. For example, it was often assumed
that the colour of the walls of a sick-room had no
effect upon the condition of the patient. It is now
known that certain mental patients are made worse
by seeing some colours, and are aided by seeing others.
The only way to avoid making mistakes of this kind
is to resort to comparison of cases in which different
factors are varied. The most satisfactory procedure
is to test by experiment, i.e. by deliberately varying
a given factor and observing what happens. In an
experiment the observer is able to control the con-
ditions in such a way that he can vary the factor he
is investigating without thereby varying other factors.
Wherever experiment is possible, hypotheses with
regard to possible causes can be tested. It is not
difficult to see that the field for experimental testing
is limited to those situations in which the observer can
deliberately arrange to initiate those changes the
results of which he wishes to observe. Just as a skil-
ful barrister, in cross-examination of a witness, asks

Cf. p. 3 above.

THE ESTIMATION OF EVIDENCE

95

those questions which are most likely to yield the
answers he wants, so a skilful experimenter arranges
those conditions the observation of which will answer
the questions constituting his problem.

Much might be said about the technique of experi-
ment. But to do so would require another small
volume. For our purposes, however, it is not im-
portant to stress the part played by experimental
investigation in the more advanced sciences. Nor
need we pause to consider the bearing of experiment
upon quantitative investigation. From the strictly
logical point of view, the most complicated scientific
experiment reveals only the same logical principles as
are exemplified in our ordinary reasonings concerning
matters of fact. In both alike what matters is that
we cannot formulate a question save on the basis of
previous knowledge ; wre must make judgments of
irrelevance, since no situation presents only those
features which are significant for our problem ; we
must analyse the situation under investigation in
order to discover its relevant likenesses to, and
differences from, other situations of the same gen-
eral nature. Those principles which control sound
generalizations concerning classes are also the
principles which he at the basis of causal investigation.

CHAPTER VI

THE GROUNDS OF OUR BELIEFS 1

‘ It is undesirable to believe a proposition when there is no
ground whatever for supposing it true.’ — Bertrand Russell

WE all commonly entertain many beliefs for which
we have little, or no, evidence. Some of these
beliefs may be baseless, but some may be capable of
being supported by sound evidence, which we could
discover if we wished. Frequently, however, we do
not know, and have never thought to inquire, what
this evidence is. When, however, a cherished belief
is challenged we may be moved to argue in its support ;
when a doubt has occurred to ourself we may seek to
remove that doubt. In seeking to resolve a doubt we
are seeking premisses from which the proposition in
question follows, or which can at least be adduced as
affording some evidence for its truth. These premisses
provide logical reasons justifying belief in a given
conclusion. Frequently it happens that the evidence
is not sufficient to imply the conclusion whilst it is
sufficient to justify the belief that the conclusion is
probably true. To say that the truth of a proposition
is more or less probable is to say that there is some
evidence (more or less strong) in its favour and no
conclusive evidence against it. To have conclusive
evidence against a proposition is to have a logical
reason for disbelieving it. The notion of believing
that so-and-so is probably true is familiar to common

1 The word 1 belief ’ is used throughout simply as short for
1 that which is believed ’.

96

THE GROUNDS OF OUR BELIEFS

97

sense. If some one said, ‘ In my opinion there will be
another great European War before 1940/ he would be
tacitly admitting that the italicized statement is not
known to be certainly true, whilst asserting his belief
that the available evidence renders its truth probable.
The reader will have no difficulty in understanding
this notion of probability. He should, however,
observe that the words ‘ probability ’, ‘ opinion ’, and
‘ belief ’, are not used with precision in ordinary con-
versation. We sometimes assert an opinion when we
have no evidence at all in favour of that opinion. In
such a case it is incorrect to say that the proposition
opined is probably true, for to say just this is to say
that there is some evidence in its favour. Whilst,
therefore, some of our beliefs may not be based upon
evidence, our belief that a given proposition is
probably true requires the assumption that there are
some grounds in its favour. Probability admits of
degrees, varying between the two extremes of certainly
true and certainly false. Thus probability is not
equivalent to mere possibility, nor improbability to
impossibility. Probability is relative to the evidence,
so that the truth of a proposition may be extremely
probable (or improbable), in face of the available
evidence, and may yet be false (or true). For example,
the evidence now available with regard to the state-
ment that there will shortly be a European War may
make its truth very probable, and yet the state-
ment may be false. Some change, not now foresee-
able, in the attitude of nations might occur.
But, since unforeseeable occurrences are necessarily
unforeseen, this mere possibility is not evidence.
It is unreasonable to entertain a strong degree of
doubt with regard to a proposition which has
7

98 LOGIC IN PRACTICE

been shown to have evidence rendering its truth
probable.

Believing must be distinguished from having
knowledge. Beliefs may be false, but what is known
cannot be false, since ‘ false knowledge ’ is a contra-
diction in terms. Further, we may have a true belief
where we do not have knowledge, for wre may enter-
tain a belief, which is in fact true, only because we
believe something else wrhich is false. For example,
a juror may truly believe that an accused prisoner is
innocent simply because the juror has taken a dislike
to the principal witness and refuses to believe he is
speaking the truth, whereas his testimony may be
correct. The juror could not then be said to know
that the accused prisoner is innocent, although his
belief would be true. A judgment which is defended
on false grounds cannot be known to be true, even
when it is in fact true, so that in believing it we should
be believing truly. We should only be in the position
of happening to believe what is the case without
knowing how it was the case. Most of what commonly
passes for knowledge is at best only opinion or belief
having a considerable degree of probability of truth.
From the practical standpoint, however, it would be
inconvenient to refuse to accept as knowledge what
has, in fact, only a high degree of probability. A high
degree of probability is often called ‘ practical
certainty ’. A reasonable man should not refrain
from acting upon a practical certainty as though it
were known to be true. In England, for instance, it
is customary for a Judge, at the trial of a person
accused of murder, to instruct the jury that an
adverse verdict need not be based upon the belief
that the guilt of the prisoner has been ‘ proved ’, but

THE GROUNDS OF OUR BELIEFS

99

upon the belief that the guilt has been established
‘ beyond reasonable doubt To be ‘ beyond reason-
able doubt ’ is to have sufficient evidence to make the
proposition in question so much more likely to be
true than to be false that we should be prepared to
act upon the supposition of its truth. Many of our
most important actions have to be performed in
accordance with beliefs of such a land. A healthy
youth acts reasonably if he prepares himself for a
career, notwithstanding the possibility that he may
die before its fruition. If, however, he sets out on
an extremely hazardous adventure, he would act
reasonably in making his will beforehand.

It is important that our beliefs should not be such
as a reasonable man would be compelled to reject.
Nevertheless, we often have to act upon a definite
belief although there is much to be said on the opposite
side. This is the case with many beliefs about
politics, about our educational policy, about our
charities. When we must act in one way or the other,
it is simply stupid to refrain from committing our-
selves to a definite belief, even though we may see
clearly what may be urged on the other side. All
that we can do is to act in accordance with that
belief which seems to us, after due thought, to be
more likely to be true than is the contrary belief.
The recognition that the other side is not negligible
may well make us more tolerant, but it should not
render us merely undecided in action.

Sometimes our beliefs are erroneous because we
have accepted an unrestricted generalization when
only a restricted one would be justified. Thus we
may hold that Frenchmen are always clear thinkers,
or that Englishmen are always honourable, or that
8

100

LOGIC IN PRACTICE

people who speak fluently do not think profoundly ;
in each case it might be that the substitution of
‘ usually ’ or ‘ very often ’ for ‘ always ’ would render
the belief justifiable. In some sciences much use is
made of a form of statement which enables us to
substitute, for such generalizations as the above, a
more precise proposition which has a better chance of
truth. An example may make the point clear. Let
us consider the unrestricted statement, Fluent sjjeakers
are not profound thinkers. As it stands this suggests
either that there is some causal connexion between
the ability to speak fluently and lack of ability to
think profoundly, or that the two characteristics
happen to be conjoined. On either alternative we
should be ready to regard fluent speaking as a sign
of superficial thinking. It would be reasonable,
however, to ask whether there is a greater proportion
of superficial thinkers among those who speak
fluently than among those who speak hesitatingly or
slowly. We must, then, consider the four classes :
(i) fluent speakers ; (ii) slow speakers ; (iii) superficial
thinkers ; (iv) profound thinkers. Let (i) be repre-
sented by X, (ii) by non-X, (iii) by non-Y, (iv) by Y.
Our problem is to discover whether the X's which are
non-Y exceed proportionally the non-Y who are also
non-X ? To solve this problem we must carry out
a statistical investigation, i.e. we must examine a
number of speakers, taken at random, divide them into
the four groups, XY, X non-Y, non-XY, non-X non-Y,
and then determine whether a higher percentage of
the X group than of the non-X group fall into the
non-Y class.1 If so, then we should be justified in

1 It is not possible here to do more than suggest the value of
precise statistical investigations. I do not wish to imply that

THE GROUNDS OF OUR BELIEFS

101

saying that fluent speakers tend not to think pro-
foundly. This statement would justify us in believing
that a fluent speaker is more likely than not to think
superficially ; it would not justify us in feeling certain
that fluency of speech must be combined with lack
of profundity.

We require knowledge, or at least true belief, for
the ordinary purposes of life. A belief is justified
when adequate evidence is adduced in its support.
Some of our beliefs, however, stand in no need of
justification, since they have consequences but no
grounds. Such beliefs may be called underived beliefs.
They are to be contrasted with derived beliefs, i.e.
beliefs capable of being based upon evidence. There
appear to be two kinds of underived beliefs : (1) beliefs
concerning sense-experience and memory ; (2) pre-
reflective beliefs of common sense. The ordinary
man does not question his belief, for instance, that he
is alive, or that he feels tired, or that he hears a loud
noise. He sees the difference between asserting ‘ I
hear a loud noise ’ and £ I hear a pistol-shot ’. He
would not regard it as reasonable to question whether
he hears a loud noise ; he would say he knows he does.
But he might admit the possibility that what he
heard was not a pistol-shot but the sound of a bursting
tyre. He might admit that the assertion That was a
pistol-shot could have been derived from the two
premisses : That was a noise of a certain kind and Only
pistol-shots make that kind of noise. A belief which
could significantly be questioned could also (if true) be
derived even if no one had in fact ever derived it. A

numerical ratios could be profitably introduced into ordinary
discussion, but merely to call attention to the wisdom of refraining
from sweeping generalizations which have not been tested.

102

LOGIC IN PRACTICE

belief which was not only underived but also unde-
rivable could not significantly be questioned, since it
would be meaningless to ask what were the grounds
of a belief which could have no grounds. The second
kind of underived beliefs may be called ‘ intuitive
beliefs provided that we remember that intuitions
may be mistaken. Examples of intuitive beliefs will
be found in the various logical principles we have
mentioned.

There are also two kinds of derived beliefs : (i) those
derived from what other people tell us, i.e. from testi-
mony ; (ii) those derived by inference from (1), (2),
or (i). These divisions cannot be sharply maintained
since (i) could be reduced to (ii). Possibly many
intuitive beliefs could be derived by inference.
Usually, however, we do not so derive them. This
fourfold division of kinds of beliefs may be con-
veniently adopted here.

What we can know directly by means of sense-
experience and memory constitutes a very small
portion of our unquestioned beliefs. This little store
of knowledge is considerably increased by accepting
testimony and by deliberate drawing of inferences
from what is thus known. Inference is indeed the
most common way of increasing our knowledge. It
is not to be suggested that testimony should be
unhesitatingly accepted, nor that our pre-reflective
beliefs should never be questioned. But doubts are
fruitful only when we are prepared to think simply in
order to discover whether our beliefs are justifiable,
and when we have some knowledge of how to set
about justifying them.

It is this problem of justification which interests
the logician. The psychologist is interested in the

THE GROUNDS OF OUR BELIEFS

103

analysis of mental attitudes and in the problem how
we come to believe or doubt something. As practical
logicians we are interested in the latter problem only
in so far as knowing why we reach the beliefs we
entertain would help us in practice not to believe or to
doubt without justification. We want to be able to
distinguish between good reasons and bad reasons,
i.e. to distinguish between an argument which is
logically sound and one which, although it convinced
us, was nevertheless unsound.

There are at least five different ways in which we
may come to hold some non-intuitive belief. (1) We
may believe a proposition because we have frequently
heard it asserted and have never thought of question-
ing it. We may even be unaware that our acceptance
is based upon what people have told us, for we may
have grown up with the beliefs. The accepted com-
monplaces of thought fall under tills head, e.g. Murder
will out. Many beliefs accepted in this way are true,
but some are not ; if we do not recognize that a
belief thus reached may, for all we know, be erroneous,
we may some day get a severe shock. (2) We may
accept a belief on the authority of a parent, or a
teacher, or a church, or some social institution. Such
acceptance presupposes the belief that the authority
is reliable. This, again, may well be the case, but
mere reliance on authority involves risk of error.
(3) Our belief in a given statement may be based upon
acceptance of the testimony of an expert. The field
of knowledge is so extensive that no one can hope to
have first-hand knowledge concerning many interest-
ing and important topics. An expert is a person who
has made a special study of a given subject and has
thus acquired competence therein. This competence

104

LOGIC IN PRACTICE

renders him reliable within the field of his study.
It would be foolish for a layman to question the
correctness of a scientific statement made by an
expert in that branch of science. For instance, a
person who has never studied the marriage customs
of primitive peoples has no good ground for believing
that polyandry is contrary to human nature if it is
the case that anthropologists have professed to
provide evidence that some tribes practise polyandry.
It is true that experts do not always agree, but their
disagreements cannot be evaluated by a layman.
Moreover, sometimes they do agree. It is well to
remember, as Bertrand Russell has pointed out,

‘ that when the experts are agreed, the opposite
opinion cannot be held to be certain and ‘ that when
they are not agreed, no opinion can be regarded as
certain by a non-expert.’ The reader may think that
no one w'ould dissent from this counsel. Nevertheless,
we all do tend to hold firmly certain views about
matters arousing our emotional interests, although
these views would be decisively rejected by all the
experts. Whilst it is reasonable to accept expert
testimony, it is foolish to allow an expert in a special
subject to dictate to us outside the limits of that
subject. At the present time there is a tendency to
allow scientists to tell us what we ought to think
about subjects in which they have no special
competence.

The three cases just discussed relate to beliefs which
are derivative, but which we have not ourselves
reached by explicit inference. The next two cases
relate to beliefs consciously accepted after a process
of questioning. We are now concerned with ways of
resolving doubt or of removing admitted ignorance.

THE GROUNDS OF OUR BELIEFS 105

Whenever we strongly entertain a belief which we
consider important, we are tempted to induce other
people to share our belief. The desire to secure agree-
ment may be so strong that we may be willing to use
any means capable of attaining our aim. Cases (4)
and (5) fall under this heading. They are to be distin-
guished by the nature of the means employed. One
way may be called the method of persuasion, the
other the method of conviction.

It is often said that the art of persuading to agree-
ment is the art of oratory. But it is to be feared that
many possess the power to persuade who do not
possess the attractive gift of oratory. Certain wiles,
however, are possessed in common by orators and by
those who make public speeches. An illustration may
make clear the method of persuasion. In October
1932, Mr. Stanley Baldwin broadcast a speech, shortly
after the resignation of certain Cabinet Ministers from
the National Government. Mr. Baldwin stated that
it w'as his intention to give the reasons why he and his
friends intended to ‘ stick to the Government ’.
Without further preamble he said :

‘ A little over a year ago the ship of state was
heading for the rocks. The skipper had to change
his course, suddenly, and many of his officers and
most of his crew deserted. It was a case of all
hands to the pumps, and I signed on with my
friends, not for six months or a year ; I signed on
for the duration, be the weather fair or foul, and I
am going to stick to the ship, whether it goes to the
bottom or gets into port, and I think the latter end
is a good deal more likely.’

This is a skilful statement. It at once induces an

106

LOGIC IN PRACTICE

attitude of acceptance. Every one would admit that
it is dastardly to desert a ship heading for the rocks,
could working at the pumps save it. Few of his
hearers would pause to ask whether the skipper and
the rest of the crew might have escaped the rocks
had they joined the deserters — probably in the boats.
A British audience in 1932 would have an emotional
attitude to the phrase ‘ signed on for the duration
since it awakens memories of sacrifices made during
the Great War. A wave of sympathy would be felt
for the brave officer who will ‘ stick to the ship ’ even
unto death. But the whole force of the argument, if
it be intended to provide good reasons for remaining in
the National Government, depends upon the sound-
ness of the comparison between the position of a
government, on one hand, and the position of the
officers and crew of a ship, on the other ; and between
the position of the National Government in 1931 and
a ship on the rocks. No training in logic is required
to enable us to see that in no single detail is there any
relevant likeness between the things compared. It
might as well be retorted that the brave officers were
those who first showed the way by springing into the
angry sea. This argument may persuade a stupid
electorate ; it cannot convince any one.

The method of conviction consists in seeking to
secure acceptance of a proposition by shoving that it
is derived from sound reasons, i.e. from evidence
adequate for its support. It has been the aim of this
book to indicate the conditions of adequate evidence.
Lack of space, as well as the incompetence of the
author, has made a full treatment impossible. Enough
has been said if the reader has been convinced of the
importance and the difficulty of clear thinking with

THE GROUNDS OF OUR BELIEFS

107

regard to the grounds of our beliefs. A pedantic
demand for the grounds of all our beliefs is not to be
encouraged. But if we do not wish to be at the
mercy of a skilful but unscrupulous persuader, we
must have some awareness of what sort of argument
can properly be adduced for any statement we are
asked to accept. It is only possible here to notice
that different sorts of statements require different
sorts of evidence. In the case of mathematical state-
ments, the evidence offered is abstract, and the form
of argument must be rigidly deductive. Statements,
in physical science can never be supported by purely
deductive reasoning, since their ultimate test is to be
found in experimental confirmation. Nevertheless,
deductive reasoning plays a considerable part. In the
case of the social sciences, such as economics, socio-
logy, and anthropology, as in political science, state-
ments should be supported by evidence obtained from
observation, aided by statistical investigation. There
is no place for pre-reflective beliefs. It is foolish to
believe that socialism is incompatible with human
nature unless we can state definitely with what
established psychological generalizations the theory of
socialism comes into conflict. In this statement
‘ capitalism ’ could equally well be substituted for
‘ socialism ’.

Finally, our beliefs about matters requiring expert
investigation must be accepted on the authority of
the experts. No training in logic makes us competent
to have opinions about physics or about religion. But
in one sense we do not have beliefs about subjects of
which we understand nothing ; we have only opinions
capable of verbal expression. To be able to say ‘ I
accept the fact that nothing can travel faster than

108

LOGIC IN PRACTICE

light ’ is quite different from the insight that this
is so. Insight comes from clear apprehension of
relevant connexions.

There is only one rehable method by which we may
increase our knowledge, or, where knowledge is im-
possible, be led to entertain reasonable behefs. The
method consists in apprehending the relevance of that
which we already know to that which we do not yet
know but are able to discover because what we already
know is significant of that which is unknowm.

REFERENCES FOR READING

IN addition to the works mentioned in the text, the following
books are recommended to those who desire to read more
on some of the topics which have been discussed :

R. H. Thouless : Straight and Crooked Thinking. (Hodder &
Stoughton.)

L. J. Russell : An Introduction to Logic. From the Standpoint
of Education. (Macmillan & Co.)

A. Sidgwick : The Use of Words in Reasoning. (A. & C. Black.)

E. A. Burtt : Principles and Problems of Right Thinking. (Harper
& Brothers.)

INDEX

About, two senses of, 43
Abstraction and analysis, 18
Affirmation and denial, 36-8
Affirmative proposition, 37
Agreement, Principle of, 91
Alternative argument, 60
proposition, 67
Ambiguity, 70 seq.

Analogy, 86, 87
Antecedent, 56

Applicative Principle, 48, 53,
62

Belief, 97 seq.
derived and underived, 101

Causation, fundamental prin-
ciples of, 90 seq.

Class :

and class-name, 15, 17
and things of a certain sort,
14, 39

grouping into a, 14 seq., 87
Compatibility and incompati-
bility, 33

Compound proposition, 56
Conjunction, 56, 57, 93
Consequent, 56
Contradiction, 35, 36, 39
Contrary, 35
Conversion, 43 seq.

Conviction, method of, 106

Datum :

and conclusion, 8
and premiss, 20

Deduction, Principle of, 24.

See Inference
Definition, 78 seq.

Dictum de omni, 50, 52, 62
Difference, Principle of, 91
Dilemma, 60

Disjunctive proposition, 57
Distribution, 44, 45, 50
Division, Logical, 80 seq.

Empirical generalization, 19
Entailing, 23, 24, 34, 50
Equivalence, 34, 43 seq.
Evidence :

and conclusion, 8
and probability, 97 seq.
circumstantial, 86 seq.
Excluded Middle, Principle of,
39, 62

Excluding from a class, Prin-
ciple of, 49, 52
Experiment, 94 seq.

Fallacy :

of begging the question, 75
of Consequent, 59
of Irrelevant conclusion, 82
of Post hoc ergo propter hoc,
93

of Special Pleading, 83
Form, of an argument, 29. See
Reasoning

Generalization, 14, 19, 20, 41
seq., 86 seq.

General property, 19

111

112 LOGIC IN

Hypothesis, and thinking, 5
seq., 85 seq.

Hypothetical argument, 59
proposition, 56

Important :
and purpose, 15 seq.
defined, 16
property, 18
Inclusion, relation of, 53
Indefiniteness, and ambiguity,
70 seq.

Independent propositions, 23
seq.

Inference, deductive and in-
ductive, 21 seq., 86 seq.
immediate, 38

Knowledge, 5, 10, 13, 20, 98

Language, and signs, 64 seq.

Middle Term, 50, 74
‘ Must ’ and formal conditions,
22

Negative proposition, 37, 42, 44
Non-Contradiction, Principle
of, 39, 62

Obversion, 38
Opinion, 97

Particular proposition, 41, 44,
46

Predicate, 43
Premiss, 21 seq.

Probability, 97

PRACTICE

Proposition :
defined, 22

seven relations between pro-
positions, 36 seq.

Question, determines thinking,
3 seq., 95

Reasoning, 10 seq.
formal aspect of, 29
formal conditions of, 22, 25,
28, 31

material conditions of, 8, 22,
24, 63

validity of, 28, 63
Reasons, 32

Reference, uniqueness of, 66
seq.

Referend, 66
Relational argument, 55
Relevance :

and intelligent thinking, 3
seq.

and significance, 8 seq., 86,
95

Significance. See Relevance
Signs, 9, 64 seq.

Simple proposition, 56
Singular proposition, 54
Skipped Intermediaries, Axiom
of, 54

Sub-contrary, 35
Sub-implication, 35
Subject-term, 43
Sufficient condition, 51
Super-implication, 35
Syllogism :
defined, 47
forms of, 47 seq.
rules of, 53

INDEX

113

Symbols :
and signs, 64
use of, 40

Symmetry of relations, 44
Thinking :

conditions of effective, 12
purposive, and idle reverie,
1 seq.

Transitivity, 53 seq.

Universal proposition, 41

Vagueness, 70 seq.

Validity, 28, 29, 53, 61

Words :

and sounds or marks, 64 seq.
descriptive, 67 seq.

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